Weighted energy method for semilinear wave equations with time-dependent damping

Abstract

Of concern is the energy decay property of solutions to wave equations with time-dependent damping. A reasonable class of damping coefficients for the framework of weighted energy methods is proposed, which contains not only the model of “effective” damping $(1+t)^{-\beta }$ $(-1\le \beta <1)$, but also non-differentiable functions with a suitable behavior at $t\rightarrow \infty$. As an application of the weighted energy estimate, global existence for the corresponding semilinear wave equation is discussed.

1 Introduction

As is well-known, the damped wave equation $\partial _t^2u-\Delta u+\partial _tu=0$ has a diffusive structure close to that of the heat equation $\partial _tv-\Delta v=0$. Actually, the long-time behavior of solutions to the damped wave equation is characterized as the ones to the heat equation. This phenomenon is so-called diffusion phenomenon and it is well-studied in the literature. In the analysis of the damped wave equation, the above diffusive structure can be seen everywhere. Particularly, the validity of “weighted energy method” may be understood as one of the greatest features of the diffusive structure of the damped wave equation. It is natural to ask what kind of generalization preserves this kind of the diffusive structure. In this paper, we focus our attention to such a problem for the wave equation with time-dependent damping coefficients.

To fix the target problem, we introduce the following initial-boundary value problem of semilinear wave equations with the time-dependent damping coefficient b(t):

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t^2u(x,t)-\Delta u(x,t)+ b(t)\partial _tu(x,t)=f\big (u(x,t)\big ), &{}(x,t)\in \Omega \times (0,T), \\ u(x,t)=0, &{}(x,t)\in \partial \Omega \times (0,T), \\ (u,\partial _tu)(x,0)=(u_0(x),u_1(x)), &{}x\in \Omega , \end{array}\right. }\nonumber \\ \end{aligned}$$

(1.1)

where $\Omega =\mathbb {R}^N$ or an exterior domain in $\mathbb {R}^N$ ($N\ge 2$) with a smooth boundary $\partial \Omega$; the boundary condition in (1.1) is ignored when $\Omega =\mathbb {R}^N$. The coefficient $b:[0,\infty )\rightarrow \mathbb {R}$ is a given positive continuous function belonging to a certain class which contains the model of the effective damping $b(t)=(1+t)^{-\beta }$ with $-1\le \beta <1$. The nonlinearlity f(u) satisfies the usual power-type condition

$$\begin{aligned} f\in C^1(\mathbb {R}), \quad f(0)\!=\!0, \!\quad |f(\xi )-f(\eta )|\le C_f\big (|\xi |+|\eta |\big )^{p-1}|\xi -\eta |\ (\xi ,\eta \in \mathbb {R})\nonumber \\ \end{aligned}$$

(1.2)

with some constants $1<p<\infty$ and $C_f\ge 0$. The pair $(u_0,u_1)$ is assumed to belong to the energy class ${\mathcal {H}}=H^1_0(\Omega )\times L^2(\Omega )$ with some additional assumption.

The main interest of this paper is the validity of “weighted energy method” to the linear ($f\equiv 0$) and semilinear ($f\not \equiv 0$) problems (1.1). In particular, we aim to propose a reasonable class of damping coefficients, which is admissible to the framework of weighted energy method.

The Cauchy problem of the usual damped wave equation

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t^2u(x,t)-\Delta u(x,t)+\partial _tu(x,t)=f\big (u(x,t)\big ), &{}(x,t)\in \mathbb {R}^N\times (0,T), \\ (u,\partial _tu)(x,0)=(u_0(x),u_1(x))&{}x\in \mathbb {R}^N \end{array}\right. } \end{aligned}$$

(1.3)

has been studied in the literature. In Matsumura [26], the fundamental properties of solutions are discussed and he found the estimates of linear solutions as follows:

$$\begin{aligned} E_{\mathbb {R}^N}(u;t)=\Vert \partial _tu\Vert _{L^2}^2+\Vert \nabla u\Vert _{L^2}^2\le C(1+t)^{-\frac{N}{2}-1}\Big (\Vert u_0\Vert _{H^1\cap L^1}^2+ \Vert u_1\Vert _{L^2\cap L^1}^2\Big ). \end{aligned}$$

Nowadays, the above inequality is called “Matsumura estimate” and it has been investigated to clarify the structure of dissipation and applied to semilinear problems (see e.g., a recent research Ikeda–Taniguchi–Wakasugi [18] and the references therein). In Hsiao–Liu [12], they proved diffusion phenomena, that is, the solution u of (1.3) satisfies

$$\begin{aligned} \Vert u(\cdot ,t)-v(\cdot ,t)\Vert _{L^2} = o\big (\Vert v(\cdot ,t)\Vert _{L^2}\big ), \quad \text {as}\ t\rightarrow \infty , \end{aligned}$$

(1.4)

where v is the solution of the following Cauchy problem of the heat equation

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t v(x,t)-\Delta v(x,t)=0, &{}(x,t)\in \mathbb {R}^N\times (0,\infty ), \\ v(x,0)=u_0(x)+u_1(x), &{} x\in \mathbb {R}^N \end{array}\right. } \end{aligned}$$

(1.5)

(see also Yan–Milani [47]). For the semilinear problem ($f\not \equiv 0$), global existence and finite time blowup of solutions to (1.3) also have been considered in the previous works. In Todorova–Yordanov [40], the problem (1.3) with compactly supported initial data is discussed. It is proved by a framework of weighted energy methods that if $1+\frac{2}{N}<p<\frac{N+2}{N-2}$ ($1+\frac{2}{N}<p<\infty$ for $N=1,2$), then the problem (1.3) possesses a global-in-time solution. On the one hand, if $f(\xi )=|\xi |^p$ and $1<p<1+\frac{2}{N}$, then the problem has a finite blowup solution for any small initial data the blowup result for the threshold case $p=1+\frac{2}{N}$ is filled in Zhang [48]. Therefore the critical exponent (threshold of p for small data global existence and small data blowup) has been determined as $p=1+\frac{2}{N}$; note that the semilinear heat equation $\partial _tu-\Delta u=u^p$ has the Fujita exponent $p=1+\frac{2}{N}$ as the critical exponent for existence and non-existence of positive solutions (see Fujita [9]). Later, in Ikehata–Tanizawa [23], the assumption for the compactness of supports of initial data has been removed, but still an exponential decay of initial data is required in the sense of the restriction

$$\begin{aligned} \int _{\mathbb {R}^N}\Big (|\nabla u_0|^2+u_1^2+u_0^2\Big )e^{\mu |x|^2}\,dx<+\infty . \end{aligned}$$

The class of initial data including polynomially decaying functions in $L^r$ $(1\le r<2)$ can be found in Nakao–Ono [27], Galley–Raugel [10], Ikehata–Ohta [22], Hayashi–Kaikina–Naumkin [11] (see also Ikeda–Inui–Okamoto–Wakasugi [15] and the references therein).

In the case of time-dependent damping, Wirth [43] studied the classification of the asymptotic profiles of the solutions to the linear problem

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t^2u(x,t)-\Delta u(x,t)+b(t)\partial _tu(x,t)=0, &{}(x,t)\in \mathbb {R}^N\times (0,\infty ), \\ (u,\partial _tu)(x,0)=(u_0(x),u_1(x)), &{}x\in \mathbb {R}^N \end{array}\right. } \end{aligned}$$

(1.6)

via the representation of solutions via the Fourier analysis. Roughly speaking, the following asymptotic profiles for the case $b(t)=(1+t)^{-\beta }$ are shown:

• (i) (Non-effective case) If $\beta \in (1,\infty )$, then the solution of (1.6) behaves like the one of wave equation without damping.

• (ii) (Effective case) If $\beta \in [-1,1)$, then the solution of (1.6) behaves like the one of the parabolic equation of the form

  $$\begin{aligned} b(t)\partial _tv(x,t)-\Delta v(x,t)=0, \quad (x,t)\in \mathbb {R}^N\times (0,\infty ). \end{aligned}$$

• (iii) (Overdamping case) If $\beta \in (-\infty ,-1)$, then the solution of (1.6) converges to a non-trivial state.

An abstract version of the effective case (ii) can be seen in Yamazaki [44, 45] via the spectral analysis with the selfadjoint operator in a Hilbert space. Note that the classification of effectiveness and non-effectiveness is written in Reissig–Wirth [33] via the validity of Stricharz estimates which is different from our terminology.

The critical exponent for the corresponding semilinear problem

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t^2u(x,t)-\Delta u(x,t)+b(t)\partial _tu(x,t)=f\big (u(x,t)\big ), &{}(x,t)\in \mathbb {R}^N\times (0,\infty ), \\ (u,\partial _tu)(x,0)=(u_0,u_1), &{}x\in \mathbb {R}^N \end{array}\right. }\nonumber \\ \end{aligned}$$

(1.7)

is also considered. The time-dependent version of energy methods with exponential weights can be found in Nishihara [28, 29], Lin–Nishihara–Zhai [24] and D’Abbicco–Lucente–Reissig [5]. In D’Abbicco [3], the framework of Ikehata–Ohta [22] has been used. The method of scaling variables (originated by Galley–Raugel [10]) can be seen in Wakasugi [41, 42]. For the study of blowup solutions with small initial data, it is discussed in D’Abbicco–Lucente [4] and in Ikeda–Sobajima–Wakasugi [17]. Detailed information for the case $b(t)=1+t$ (that is, $\beta =-1$) is analysed in Ikeda–Inui [14]. If $b(t)=\mu (1+t)^{-1}$ (the case $\beta =1$), then the structures of the wave and heat equations appear. Therefore this is the threshold case for the In this case, it is expected that the critical exponent seems to depend on the constant $\mu$, but it is still remained as open problems. We do not enter the detail of this case (for the detail, a recent research D’Abbicco [6] and the references therein). More general class of abstract evolution equations (in a Hilbert space) of the form

$$\begin{aligned} \frac{d^2u}{dt^2}+c(t)Au+b(t)\frac{du}{dt}=0, \quad t\ge 0 \end{aligned}$$

is also treated in Yamazaki [46] under a weaker condition on b (and also c) compared with Wirth [43].

On the other hand, the problem of the semilinear damped wave equation

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t^2u(x,t)-\Delta u(x,t)+\partial _tu(x,t)=f\big (u(x,t)\big ), &{}(x,t)\in \Omega \times (0,T), \\ u(x,t)=0, &{}(x,t)\in \partial \Omega \times (0,T), \\ (u,\partial _tu)(x,0)=(u_0(x),u_1(x)), &{}x\in \Omega \end{array}\right. } \end{aligned}$$

(1.8)

in an exterior domain $\Omega$ (with a smooth boundary $\partial \Omega$) is rather delicate because of the reflection at the boundary. Ono [31] discussed global existence for (1.8) with (1.2) via the result of Dan–Shibata [7]. In Ikehata [20, 21], the global existence for the 2-dimensional case is proved via the weighted energy method when $2<p<\infty$. The blowup result for small solutions (when $1<p\le 1+\frac{2}{N}$) can be found in Ogawa–Takeda [30], Fino–Ibrahim–Wehbe [8] and Ikeda–Sobajima [16]. Recently, in Sobajima–Wakasugi [36], a framework of weighted energy methods for (1.8) (with space-dependent damping) is proposed, which is applicable to solutions with the initial data satisfying

$$\begin{aligned} I_{\Omega }^\lambda (u_0,u_1) =\int _{\Omega }\Big (|\nabla u_0|^2+u_1^2+u_0^2\Big ){}\langle {}x{}\rangle {}^{2\lambda }\,dx<+\infty \end{aligned}$$

with $\lambda \in [0,\frac{N}{2})$, where we use the notation ${}\langle {}x{}\rangle {}=\sqrt{1+|x|^2}$ for $x\in \mathbb {R}^N$. As an application of the framework in [36], global existence for (1.8) ($f\not \equiv 0$) is discussed in Sobajima [34]. More precisely, it is shown in [34] that if p satisfies

$$\begin{aligned} p(N,\lambda )=1+\frac{4}{N+2\lambda }\le p<p_*(N), \end{aligned}$$

then the corresponding solution of (1.8) exists uniquely when the initial data satisfy $I_{\Omega }^\lambda (u_0,u_1)\ll 1$ with $\lambda \in [0,\frac{N}{2})$ (the one-dimensional case is filled in Sobajima–Wakasugi [37]). In the recent paper Sobajima [35], an alternative approach of energy method with time-dependent dissipation is proposed, which enables us to find out the asymptotic expansion of linear solutions to the wave equation with time-dependent damping.

The goal of the present paper is to combine the framework of Sobajima–Wakasugi [36] and Sobajima [35]. More precisely, we aim to establish a framework of weighted energy methods for a (possibly wider) class of time-dependent damping, which is applicable to the initial data having a polynomial decay.

To state our result precisely, we give the definition of local-in-time and global-in-time solutions to (1.1) which mainly focused in the present paper is the following.

Definition 1.1

Let $T\in (0,\infty )$ and $b\in C([0,\infty ))$. The function $u:\Omega \times [0,T)\rightarrow \mathbb {R}$ is called a local-in-time weak solution of (1.1) in (0, T) if $u\in C^1([0,T);L^2(\Omega )) \cap C([0,T);H^1_0(\Omega ))$, $f(u)\in C([0,T);L^2(\Omega ))$ and ${\mathcal {U}}=(u,\partial _tu)$ satisfies

$$\begin{aligned} {\mathcal {U}}(t) = e^{t{\mathcal {L}}}(u_0,u_1) + \int _0^t e^{(t-s){\mathcal {L}}} {\mathcal {N}}\big (s,{\mathcal {U}}(s)\big )\,ds, \quad t\in [0,T), \end{aligned}$$

where $\{e^{t{\mathcal {L}}}\}_{t\ge 0}$ is the $C_0$-semigroup in ${\mathcal {H}}(=H^1_0(\Omega )\times L^2(\Omega ))$ generated by ${\mathcal {L}}(u,v)=(v,\Delta u)$ endowed with domain $D({\mathcal {L}})=(H^2(\Omega )\cap H_0^1(\Omega ))\times H^1_0(\Omega )$, and ${\mathcal {N}}(s,(u,v))=(0,-b(s)v+f(u))$.

Moreover, the function $u:\Omega \times [0,\infty )\rightarrow \mathbb {R}$ is called a global-in-time weak solution of (1.1) if the restriction $u|_{\Omega \times [0,T)}$ is a local-in-time weak solution of (1.1) in (0, T) for every $T\in (0,\infty )$.

The result for existence and uniqueness of local-in-time weak solutions is well-known via the Sobolev embedding $H_0^1(\Omega )\hookrightarrow L^{2p}(\Omega )$ (see e.g, Cazenave–Haraux [2] and Pazy [32]).

Proposition 1.1

Put $p_*(1)=p_*(2)=\infty$ and $p_*(N)=\frac{N}{N-2}$ for $N\ge 3$. Assume that f satisfies (1.2) with $1<p<p_*(N)$ and $b\in C([0,\infty ))$. Then for every $R>0$, there exists a positive constant $T>0$ such that the following holds: if $(u_0,u_1)\in {\mathcal {H}}$ with $\Vert u_0\Vert _{H^1_0(\Omega )}+\Vert u_1\Vert _{L^2(\Omega )}\le R$, then there exists a unique local-in-time weak solution u in (0, T).

Here we introduce our definition of the class ${\mathcal {D}}_{\textrm{diff}}$ of damping coefficients.

Definition 1.2

Let $b\in C([0,\infty ))$ be positive-valued. Define the sets ${\mathcal {D}}_{*}$, ${\mathcal {D}}_{\textrm{over}}$ and ${\mathcal {D}}_{\textrm{diff}}$ as follows;

• (i) $b\in {\mathcal {D}}_{*}$ if there exists $\zeta \in C^1([0,\infty ))$ such that $\lim \nolimits _{t\rightarrow \infty }\zeta '(t)=0$ and $\lim \nolimits _{t\rightarrow \infty }\big (b(t)\zeta (t)\big )=1$;

• (ii) $b\in {\mathcal {D}}_{\textrm{over}}$ if $b^{-1}\in L^1(0,\infty )$;

• (iii) ${\mathcal {D}}_{\textrm{diff}} ={\mathcal {D}}_*\setminus {\mathcal {D}}_{\textrm{over}}$.

Remark 1.1

Of course the model $b(t)=(1+t)^{-\beta }$ $(-1\le \beta <1)$ belongs to the class ${\mathcal {D}}_{\textrm{diff}}$ via the choice $\zeta _\beta (t)=(1+t)^{\beta }$. Moreover, one can see that the class ${\mathcal {D}}_{\textrm{diff}}$ also contains the coefficient b satisfying

$$\begin{aligned} \lim _{t\rightarrow \infty }\big (b(t)\zeta _\beta (t)\big ) = \lim _{t\rightarrow \infty }\big (b(t)(1+t)^{\beta }\big ) =0. \end{aligned}$$

Remark 1.2

The semilinear problem (1.7) has been mainly considered with b with the following assumption

• (i) $b(t)>0$ for any $t\ge 0$,

• (ii) b(t) is monotone, and $t b(t)\rightarrow +\infty$ as $t\rightarrow \infty$,

• (iii) $\frac{1}{(1+t)^2b(t)}\in L^1(0,\infty )$,

• (iv) $b\in C^3([0,\infty ))$ and $|b^{(k)}(t)|\le C(1+t)^{-k}b(t)$ for any $k=1,2,3$,

• (v) there exists $\ell <1$ such that $tb'(t)\le \ell b(t)$ for any $t\ge 0$.

The above conditions are so-called effective essentially proposed in Wirth [43]. There are important to find a suitable estimate for linear solutions (like the Matsumura estimate) to apply the nonlinear term as an inhomogeneous term (see D’Abbicco–Lucente–Reissig [5] and the subsequent papers). The coefficient b in this class also belongs to ${\mathcal {D}}_*$ via the computation with (ii) and (iv) with $k=1$; one can choose $\zeta =b^{-1}$ with

$$\begin{aligned} \zeta '(t)=-\frac{t b'(t)}{b(t)}\cdot \frac{1}{tb(t)}\rightarrow 0\quad (\text {as}\ t\rightarrow \infty ). \end{aligned}$$

Remark 1.3

The special class of the damping $b\in {\mathcal {D}}_*$ with the choice $\zeta =b^{-1}$ has been discussed in Ikeda–Sobajima–Wakasugi [17]. In this case, one requires the differentiability for b which is essential to employ the technique of scaling variables. The definition of ${\mathcal {D}}_*$ enables us to choose a certain non-differentiable damping. The assumption for b in Yamazaki [46] is also covered. The damping of the form

$$\begin{aligned} b(t)=\frac{(\log (2+t))^\varepsilon }{1+t}(1+\nu \sin (\log (2+t))), \quad (\varepsilon >0,\ |\nu |<1) \end{aligned}$$

is applicable to the framework of Yamazaki [46] and also to our framework with $\zeta =b^{-1}$. Our condition allows us to choose the (slightly generalized) damping

$$\begin{aligned} b(t)=\frac{(\log (2+t))^\varepsilon }{1+t}(1+\nu \sin (\log (2+t)))(1+{\tilde{b}}(t)), \quad (\varepsilon >0,\ |\nu |<1) \end{aligned}$$

with ${\tilde{b}}(t)=o(1)$ as $t\rightarrow \infty$ even if ${\tilde{b}}(t)$ rapidly oscillates like $\frac{|\sin (t^k)|}{1+t}$ ($k>0$).

The main novelty of this paper is the following. If the damping b belongs to the class ${\mathcal {D}}_{\textrm{diff}}$, then the following theorem asserts that the linear wave equation with the damping b has a kind of diffusive structure. The statement is formulated in the weighted energy inequality.

Theorem 1.2

Assume $b\in {\mathcal {D}}_{\textrm{diff}}$ and $f\equiv 0$. Then for every $\lambda \in [0,\frac{N}{2})$, there exists a positive constant $C_\lambda >0$ such that for every $(u_0,u_1)\in {\mathcal {H}}$ satisfying

$$\begin{aligned} I_{\Omega }^\lambda (u_0,u_1)=\int _\Omega \Big (u_1^2+|\nabla u_0|^2+u_0^2\Big ){}\langle {}x{}\rangle {}^{2\lambda }\,dx<+\infty , \end{aligned}$$

one has

$$\begin{aligned} (1+B(t))\int _\Omega \Big (|\nabla u|^2+(\partial _tu)^2\Big )W^{\lambda }\,dx + \int _\Omega u^2W^\lambda \,dx \le C_\lambda I_{\Omega }^\lambda (u_0,u_1), \end{aligned}$$

(1.9)

where

$$\begin{aligned} W(x,t)=B(t)+{}\langle {}x{}\rangle {}^2, \quad B(t)=\int _0^t\frac{1}{b(s)}\,ds. \end{aligned}$$

Remark 1.4

Basically, we can choose $\zeta$ in the definition of ${\mathcal {D}}_{\textrm{diff}}$ as a smooth function with a good property such as the derivative of a well-behaved function. Of course we can replace B(t) in the estimate (1.9) with

$$\begin{aligned} \int _{0}^t\zeta (s)\,ds,\quad t\ge 0. \end{aligned}$$

Remark 1.5

The linear solution u in Theorem 1.2 satisfies the energy decay

$$\begin{aligned} E_\Omega (u;t)=\int _\Omega \Big (|\nabla u|^2+(\partial _tu)^2\Big )\,dx \le C\big (1+B(t)\big )^{-\lambda -1}. \end{aligned}$$

In particular, if $b(t)=1+t$ (the threshold case for overdamping), then one has the logarithmic decay $E_\Omega (u;t)\le C(1+\log (1+t))^{-\lambda -1}$. In the overdamping case $b\in {\mathcal {D}}_*\cap {\mathcal {D}}_{\textrm{over}}$, Theorem 1.2 also provides an estimate for the weighed energy, but does not give new information.

We can also show global existence of weak solutions to (1.1) via the framework of the weighted energy method in Theorem 1.2.

Theorem 1.3

Assume $b\in {\mathcal {D}}_{\textrm{diff}}$ and (1.2) with $p(N,\lambda )\le p<p_*(N)$ for some $\lambda \in [0,\frac{N}{2})$. Then there exists a positive constant $\delta _*>0$ such that if

$$\begin{aligned} I_{\Omega }^\lambda (u_0,u_1)=\int _\Omega \Big (u_1^2+|\nabla u_0|^2+u_0^2\Big ){}\langle {}x{}\rangle {}^{2\lambda }\,dx\le \delta _*, \end{aligned}$$

then there exists a unique global-in-time weak solution u of (1.1). Moreover, there exists a positive constant $C>0$ such that for every $t\ge 0$,

$$\begin{aligned} (1+B(t))\int _\Omega \Big (|\nabla u|^2+(\partial _tu)^2\Big )W^{\lambda }\,dx + \int _\Omega u^2W^\lambda \,dx \le CI_{\Omega }^\lambda (u_0,u_1). \end{aligned}$$

From the viewpoint of global existence of (1.1) (with well-behaved damping b) for small initial data, it is proved in D’Abbicco–Lucente–Reissig [5] that if $\Omega =\mathbb {R}^N$ and $1+\frac{2}{N}<p<p_*(N)$, then the problem (1.1) possesses a non-trivial global-in-time weak solution. In contrast, even if the damping is not so regular, Theorem 1.3 asserts that the existence of a suitable auxiliary function $\zeta$ immediately provides a global existence result with $1+\frac{2}{N}<p<p_*(N)$ for exterior domains. The statement is as follows.

Corollary 1.4

Assume $b\in {\mathcal {D}}_{\textrm{diff}}$ and (1.2) with $1+\frac{2}{N}<p<p_*(N)$. Then there exists a positive constant $\delta _{**}>0$ such that if

$$\begin{aligned} I_{\Omega }^{\frac{N}{2}}(u_0,u_1)=\int _\Omega \Big (u_1^2+|\nabla u_0|^2+u_0^2\Big ){}\langle {}x{}\rangle {}^{N}\,dx\le \delta _{**}, \end{aligned}$$

then there exists a unique global-in-time weak solution of (1.1).

Remark 1.6

For the overdamping case $b\in {\mathcal {D}}_*\cap {\mathcal {D}}_{\textrm{over}}$, global existence of small solutions to (1.1) with $1<p<p_*(N)$ has been proved in Ikeda–Wakasugi [19]. Our proof can be modified for this cases. This is a consequence of the following fact (with $\lambda =0$)

$$\begin{aligned} (1+B(t))^{1-\frac{N}{4}(p-1)}\le {\left\{ \begin{array}{ll} 1 &{}\text {if}\ 1+\frac{4}{N}\le p<p_*(N), \\ \big (1+\Vert b^{-1}\Vert _{L^1(0,\infty )}\big )^{1-\frac{N}{4}(p-1)} &{}\text {if}\ 1< p<1+\frac{4}{N}. \end{array}\right. } \end{aligned}$$

Here we briefly show our idea of the treatment of damping terms. In the case of the usual damped wave equation $\partial _t^2u-\Delta u+\partial _tu=0$, the equation can be written by the alternative form $\partial _t(\partial _tu+u)=\Delta u$. Although this modification is trivial, one can directly see important information of the asymptotic behavior in (1.5) in the above form. If we move to the problem with time dependent damping b(t), as an experience in Sobajima [35, Sect. 3] (written in an abstract formulation), we can find the alternative form

$$\begin{aligned} \partial _t(m(t)\partial _tu+u)=m(t)\Delta u \end{aligned}$$

with a multiplier m(t) which is required to be positive and satisfy the ordinary differential equation

$$\begin{aligned} m'(t)+1=b(t)m(t); \end{aligned}$$

(1.10)

note that the equation (1.10) already appears in Lin–Nishihara–Zhai [25] to study blowup phenomena. Note that existence of positive solutions to (1.10) is verified when $b\in {\mathcal {D}}_*$. Applying this consideration, we can see that the semilinear equation in (1.1) is rewritten as the alternative form

$$\begin{aligned} \partial _t\big (m(t)\partial _tu+u\big ) = m(t)\big (\Delta u+f(u)\big ). \end{aligned}$$

(1.11)

This form plays a crucial role to apply an energy method, where the damping coefficient depends on t. In fact, the multiplier m(t) is effectively used in [35] to derive the energy estimates via the use of the following auxiliary functional

$$\begin{aligned} \int _{\mathbb {R}^N}\big (2m(t)\partial _tu+u\big )u\,dx. \end{aligned}$$

We can see that all important quantities for the energy estimate come from the behavior of the multiplier m(t) (not directly from b(t)). Adopting the weighted energy method in Sobajima–Wakasugi [36, 38], one can expect that functionals of the form

$$\begin{aligned} \int _{\mathbb {R}^N}\big (2m(t)\partial _tu+u\big )u\Phi ^{2\delta -1}\,dx \end{aligned}$$

seem to be reasonable to carry out the weighted energy methods with time-dependent damping b(t), where $\Phi$ is a suitable function related to the parabolic equation $\partial _tv=m(t)\Delta v$. Since the above structure is independent of the domain, we can also address the initial-boundary value problem (1.1).

The present paper is originated as follows. In Sect. 2, we collect the fundamental facts (supersolutions of the linear heat equation and the Sobolev inequalities) to consider the problem (1.1). In Sect. 3, we explain how to derive the weighted energy estimates for the linear problem with inhomogeneous terms. In Sect. 4, the energy estimate (obtained in Sect. 3) is applied to discuss the semilinear problem (1.1).

2 Preliminaries

In this section, we collect some basic but important tools for the linear and nonlinear problems (1.1).

2.1 A supersolution of the heat equation

First we introduce a supersolution of the heat equation

$$\begin{aligned} \partial _t\Phi (x,t)-\Delta \Phi (x,t)\ge 0, \quad (x,t)\in \mathbb {R}^N\times (0,\infty ), \end{aligned}$$

which has been used to construct a weight function in the energy method for the damped wave equation (see also Sobajima–Wakasugi [36] or Sobajima [34]). We use a refined version of weight functions in Sobajima–Wakasugi [39].

Definition 2.1

Let $\lambda \in [0,\frac{N}{2})$. Define

$$\begin{aligned} \beta _\lambda =\frac{2N\lambda }{N+2\lambda }\in \left[ \lambda , \frac{N}{2}\right) , \quad \gamma _\lambda =\frac{N+2\lambda }{4}\in \left( \beta _\lambda , \frac{N}{2}\right) \end{aligned}$$

and

$$\begin{aligned} \Phi _{\lambda }(x,t)=t^{-\beta _\lambda }\varphi _{\lambda }\left( \frac{|x|^2}{4t}\right) , \quad \varphi _\lambda (z)=e^{-z}M(\gamma _\lambda -\beta _\lambda ,\gamma _\lambda ;z), \end{aligned}$$

where M(a, c; z) denotes the Kummer confluent hypergeometric function

$$\begin{aligned} M(a,c;z)=\sum _{n=0}^\infty \frac{(a)_n}{(c)_n}\frac{z^n}{n!}, \quad z\ge 0 \end{aligned}$$

with the Pochhammer symbol defined by $(d)_0 = 1$ and $(d)_n=\sum _{k=1}^n(d+k-1)$ for $n \in \mathbb {N}$ (see also Beals–Wong [1]).

The detail of $\Phi _\lambda$ is slightly different from supersolutions in Sobajima–Wakasugi [39] but essentially equivalent to that. The important properties of $\Phi _\lambda$ is collected in the following lemma.

Lemma 2.1

Let $\Phi _\beta$ be as in Definition 2.1 and set

$$\begin{aligned} \Psi (x,t)=t+\frac{|x|^2}{4}, \quad (x,t)\in \mathbb {R}^N\times [0,\infty ). \end{aligned}$$

Then the following assertions hold:

• (i) There exists a positive constant $\eta _\lambda >0$ such that

  $$\begin{aligned} \frac{\partial _t\Phi _\lambda (x,t)-\Delta \Phi _\lambda (x,t)}{\Phi _\lambda (x,t)} \ge \frac{\eta _\lambda }{\Psi (x,t)}, \quad (x,t)\in \mathbb {R}^N\times (0,\infty ). \end{aligned}$$

• (ii) There exist two positive constants $c_\lambda >0$ and $C_\lambda >0$ such that

  $$\begin{aligned} \frac{c_\lambda }{\Psi (x,t)^{\beta _\lambda }} \le \Phi _\lambda (x,t) \le \frac{C_\lambda }{\Psi (x,t)^{\beta _\lambda }}, \quad (x,t)\in \mathbb {R}^N\times (0,\infty ). \end{aligned}$$

• (iii) There exists a positive constant ${\widetilde{C}}_\lambda$ such that

  $$\begin{aligned} \frac{|\partial _t\Phi _\lambda (x,t)|}{\Phi _\lambda (x,t)} \le \frac{{\widetilde{C}}_\lambda }{\Psi (x,t)}, \quad (x,t)\in \mathbb {R}^N\times (0,\infty ). \end{aligned}$$

2.2 Some functional inequalities

The following inequality is one of important tools to find out a “good term” in the calculation of energy methods via integration by parts.

Lemma 2.2

[34] Let D be a bounded domain in $\mathbb {R}^N$ with a smooth boundary. Assume that $\Theta \in C^2(D)$ is positive and $\delta \in (0,\frac{1}{2})$. Then for every $v\in H^2(D)\cap H^1_0(D)$,

$$\begin{aligned} \int _D \frac{v\Delta v}{\Theta ^{1-2\delta }}\,dx \le \frac{\delta }{1-\delta }\int _D \frac{|\nabla v|^2}{\Theta ^{1-2\delta }}\,dx + \frac{1-2\delta }{2} \int _D \frac{v^2\Delta \Theta }{\Theta ^{2-2\delta }}\,dx. \end{aligned}$$

The next identity is nothing but the special case of the Gagliargo–Nirenberg inequality. The following modification is an adjusted version for weighted energy methods.

Lemma 2.3

Let D be a bounded domain in $\mathbb {R}^N$ with a smooth boundary. Let $\tau >0$ and $\mu >0$ be fixed. If $1<p<\infty$ (for $N=1,2$) or $1<p\le \frac{N+2}{N-2}$ (for $N\ge 3$), then there exists a positive constant $K_{\mu ,p}>0$ (independent of D) such that for every $v\in H_0^1(D)$,

$$\begin{aligned} \int _D|v|^{p+1}\Psi (x,\tau )^\mu \,dx&\le \frac{K_{\mu ,p}}{ {\tau ^{\frac{\mu }{2}(p-1)}} } \left( \int _Dv^2\Psi (x,\tau )^\mu \,dx \right) ^{\frac{p+1}{2}-\frac{N}{4}(p-1)} \\&\quad \times \left( \int _D\left( |\nabla v|^2+\frac{v^2}{\Psi (x,\tau )}\right) \Psi (x,\tau )^\mu \,dx \right) ^{\frac{N}{4}(p-1)}, \end{aligned}$$

where $\Psi (x,\tau )=\tau +\frac{|x|^2}{4}$.

3 A weighted energy method for linear problem

In this section, we consider the inhomogeneous problem of the damped wave equation

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t^2w(x,t)-\Delta w(x,t) + b(t)\partial _tw(x,t)=g(x,t), &{} (x,t)\in D\times (0,T),\qquad \\ w(x,t)=0, &{} (x,t)\in \partial D\times (0,T),\qquad \\ (w,\partial _tw)(x,0)=(w_0,w_1), &{}x\in D,\qquad \end{array}\right. } \end{aligned}$$

(3.1)

where $D\subset \mathbb {R}^N$ is a bounded domain with a smooth boundary $\partial D$. Here we only assume that

$$\begin{aligned} (w_0,w_1)\in H^1_0(D)\times L^2(D), \quad g\in C([0,T];L^2(D)). \end{aligned}$$

(3.2)

Existence of the energy solution of (3.1) is also well-known (see e.g., Ikawa [13]). Here we focus our attention to estimates for the functional

$$\begin{aligned} \int _D\Big (\big (\partial _tw(x,t)\big )^2+|\nabla w(x,t)|^2\Big )W(x,t)^\lambda \,dx, \end{aligned}$$

(3.3)

where W is given in Theorem 1.2. Since several properties of solutions to (3.1) (in this section) will be applied to the solution of the semilinear problem (1.1), that is, $w=u$ and $g=f(u)$, it is necessary to discuss estimates for the weighted energy (3.3) with constants independent of D.

Now we explain our strategy of weighted energy methods. Basically, we know that the usual energy equality holds:

$$\begin{aligned} \frac{d}{dt} \left( \Vert \partial _tw\Vert _{L^2(D)}^2+\Vert \nabla w\Vert _{L^2(D)}^2\right) + 2b(t)\Vert \partial _tw\Vert _{L^2(D)}^2 = 2b(t)\int _D (\partial _tw)g\,dx \end{aligned}$$

which is the fundamental property of solutions to the wave equation. Additionally, from the viewpoint (1.11), the equation in (3.1) can be reformulated as

$$\begin{aligned} \partial _t\big (m(t)\partial _tw+w\big )=m(t)\Delta u+m(t)g, \quad (x,t)\in \Omega \times (0,T) \end{aligned}$$

(3.4)

via the use of the multiplier m(t) satisfying

$$\begin{aligned} m'(t)+1=b(t)m(t), \quad t\ge 0; \end{aligned}$$

(3.5)

existence of a suitable multiplier can be seen in Lemma 3.2 under the assumption $b\in {\mathcal {D}}_*$.

3.1 An auxiliary function describing time-scale

We first construct an auxiliary function m(t) which is reasonable to proceed the energy method for the damped wave equations with time-dependent damping. Now we consider the ordinary differential equation (3.5). The existence of positive solutions to (3.5) is characterized as follows.

Lemma 3.1

Let $b\in C([0,\infty ))$. The following assertions are equivalent:

• (i) $\displaystyle \int _0^\infty \exp \Big (-\int _0^sb(r)\,dr\Big )\,ds<+\infty$.

• (ii) there exists a positive solution m(t) of (3.5).

In this case, the minimal solution m(t) of (3.5) can be written by

$$\begin{aligned} m(t)=\int _t^\infty \exp \Big (-\int _t^sb(r)\,dr\Big )\,ds, \quad t\ge 0. \end{aligned}$$

Proof

The solution of initial-value problem of (3.5) can be computed as

$$\begin{aligned} m(t)=\exp \Big (\int _0^tb(r)\,dr\Big ) \left[ m(0) -\int _0^t \exp \Big (-\int _0^sb(r)\,dr\Big )\,ds\right] . \end{aligned}$$

This immediately yields the equivalence of (i) and (ii). $\square$

Lemma 3.2

Assume that $b\in {\mathcal {D}}_*$. Then there exists a unique minimal positive solution m(t) of (3.5). Moreover, one has

$$\begin{aligned} \lim _{t\rightarrow \infty }m'(t)=0, \quad \lim _{t\rightarrow \infty }\big (b(t)m(t)\big )=1. \end{aligned}$$

(3.6)

In particular, there exist positive constants $B_0$ and $B_1$ such that for every $t\ge 0$,

$$\begin{aligned} B_0\le b(t)m(t)\le B_1. \end{aligned}$$

Remark 3.1

Well-definedness of m(t) can be found in Ikeda–Sobajima–Wakasugi [17] under the condition

$$\begin{aligned} \liminf _{t\rightarrow \infty }\Big (\frac{|b'(t)|}{b(t)^2}\Big )<1{.} \end{aligned}$$

A similar consideration provides the existence of positive solution of (3.5) under a weaker condition

$$\begin{aligned} \limsup _{t\rightarrow \infty }\zeta '(t)< \liminf _{t\rightarrow \infty }\big (b(t)\zeta (t)\big ). \end{aligned}$$

Remark 3.2

Lemma 3.2 asserts that $b\in {\mathcal {D}}_*$ if and only if $b\in C([0,\infty ))$ admits a minimal positive solution m(t) of (3.5) satisfying (3.6) (we can take $\zeta =m$). In this sense, we can assume the existence of m with (3.6) instead of that of $\zeta$ without loss of generality.

Proof of Lemma 3.2

Let $\varepsilon \in (0,1)$ be arbitrary. We see from $b\in {\mathcal {D}}_*$ that there exists $t_\varepsilon \ge 0$ such that for every $t\ge t_\varepsilon$,

$$\begin{aligned} |\zeta '(t)|<\varepsilon , \quad 1-\varepsilon<b(t)\zeta (t)<1+\varepsilon . \end{aligned}$$

These give that $\zeta (t)\le \zeta (t_\varepsilon )+\varepsilon (t-t_\varepsilon )$ and $b(t)>\frac{1-\varepsilon }{\zeta (t)}$ and therefore for every $s\ge t_\varepsilon$

$$\begin{aligned} \exp \left( -\int _0^sb(r)\,dr\right)&\le \exp \left( -\int _{t_\varepsilon }^s\frac{1-\varepsilon }{\zeta (r)}\,dr\right) \\&\le \exp \left( -\int _{t_\varepsilon }^s\frac{1-\varepsilon }{\zeta (t_\varepsilon )+\varepsilon (r-t_\varepsilon )}\,dr\right) \\&= \left( \frac{\zeta (t_\varepsilon )}{\zeta (t_\varepsilon )+\varepsilon (s-t_\varepsilon )}\right) ^{\frac{1-\varepsilon }{\varepsilon }}. \end{aligned}$$

Choosing $\varepsilon <\frac{1}{2}$, that is, $\frac{1-\varepsilon }{\varepsilon }>1$, we have the convergence of the integral in Lemma 3.1(i) which provides the existence of the minimal positive solution m(t) to (3.5).

Now we show $\lim _{t\rightarrow \infty }b(t)m(t)=1$. Define

$$\begin{aligned} m_*(t)=\int _t^\infty \exp \left( -\int _t^s\frac{1+\varepsilon }{\zeta (r)}\,dr\right) \,ds, \quad m^*(t)=\int _t^\infty \exp \left( -\int _t^s\frac{1-\varepsilon }{\zeta (r)}\,dr\right) \,ds. \end{aligned}$$

Then clearly, we have for every $t\ge t_\varepsilon$,

$$\begin{aligned} m_*(t)\le m(t)\le m^*(t). \end{aligned}$$

On the one hand, integration by part implies

$$\begin{aligned} m^*(t)=\frac{\zeta (t)}{1-\varepsilon }+ \int _t^\infty \frac{\zeta '(t)}{1-\varepsilon }\exp \left( -\int _t^s\frac{1-\varepsilon }{\zeta (r)}\,dr\right) \,ds \le \frac{\zeta (t)}{1-\varepsilon } + \frac{\varepsilon }{1-\varepsilon }m^*(t). \end{aligned}$$

This yields $m^*(t)\le \frac{\zeta (t)}{1-2\varepsilon }$. Similarly, we also have $m_*(t)\ge \frac{\zeta (t)}{1+2\varepsilon }$. Consequently, we deduce

$$\begin{aligned} \frac{1-\varepsilon }{1+2\varepsilon } \le \frac{b(t)\zeta (t)}{1+2\varepsilon } \le b(t)m(t)\le \frac{b(t)\zeta (t)}{1-2\varepsilon } \le \frac{1+\varepsilon }{1-2\varepsilon }\quad (t\ge t_\varepsilon ) \end{aligned}$$

and therefore we obtain $\lim _{t\rightarrow \infty }b(t)m(t)=1$. The other limit immediately verified via the equation $m'+1=bm$. The proof is complete. $\square$

Definition 3.1

Let $b\in {\mathcal {D}}_*$ and let m be given in Lemma 3.2. Define two functions M(t) and M(a; t) (for the parameter $a>0$) as

$$\begin{aligned} M(t)=\int _0^tm(s)\,ds, \quad M(a;t)=a+M(t) \quad (t\ge 0). \end{aligned}$$

Lemma 3.3

Let $b\in {\mathcal {D}}_*$ and let m be given in Lemma 3.2. For every $\varepsilon >0$, there exists $a_\varepsilon >0$ such that if $a\ge a_\varepsilon$ then

$$\begin{aligned} \sup _{0\le t<\infty }\frac{m(t)^2}{M(a;t)}\le \varepsilon . \end{aligned}$$

Proof

By Lemma 3.2, there exists a positive constant $t_*>0$ such that $|m'(t)|\le \frac{\varepsilon }{4}$ when $t\ge t_*$. Then we have for every $t\ge t_*$,

$$\begin{aligned} m(t)^2-m(t_*)^2 =2\int _{t_*}^tm(s)m'(s)\,ds \le \frac{\varepsilon }{2}\int _{t_*}^tm(s)\,ds \le \frac{\varepsilon }{2}M(t). \end{aligned}$$

This implies that if $a\ge 2\varepsilon ^{-1}\Vert m\Vert _{L^\infty (0,t_*)}^2$, then we have the desired inequality. $\square$

3.2 A supersolution of $\partial _t\Phi =m(t)\Delta \Phi$

To derive weighted energy estimates for (3.1), we will employ the following idea of weighted estimates for the parabolic equation $\partial _tv=m(t)\Delta v$, which appears in [36] when $b\equiv 1$: By integration by parts, we formally see that for every positive-valued function $\Phi \in C^2({\overline{D}}\times [0,\infty ))$,

$$\begin{aligned} \frac{d}{dt} \int _{D}\frac{v^2}{\Phi }\,dx&=2m(t)\int _{D}\frac{v\Delta v}{\Phi }\,dx-\int _{D}\frac{v^2\partial _t\Phi }{\Phi ^2}\,dx \\&= - 2m(t)\int _{D}\left| \nabla \left( \frac{v}{\Phi }\right) \right| ^2\Phi \,dx -\int _{D}\frac{v^2}{\Phi ^2}\big (\partial _t\Phi -m(t)\Delta \Phi \big )\,dx. \end{aligned}$$

In view of the above equality, the construction of a positive supersolution to $\partial _t\Phi =m(t)\Delta \Phi$ provides the uniform estimate for $\int _Dv^2\Phi ^{-1}\,dx$ which can be understood as a weighted $L^2$-estimate. To proceed the strategy explained above, we introduce a supersolution $\Phi _{\lambda ,a}$ of $\partial _t\Phi =m(t)\Delta \Phi$ by the trick in Sobajima–Wakasugi [38].

Definition 3.2

Let $a>0$ and let $\Phi _\lambda$ and M(a; t) be as in Definitions 2.1 and 3.1, respectively. Define

$$\begin{aligned} \Phi _{\lambda ,a}(x,t)&=\Phi _{\lambda }\big (x,M(a;t)\big ), \quad (x,t)\in \mathbb {R}^N\times (0,\infty ). \end{aligned}$$

The following lemma immediately follows from Lemma 2.1.

Lemma 3.4

Let $\Phi _{\lambda ,a}$ be as in Definition 3.2 and set

$$\begin{aligned} \Psi _a(x,t)=M(a,t)+\frac{|x|^2}{4},\quad (x,t)\in \mathbb {R}^N\times [0,\infty ). \end{aligned}$$

Then the following assertions hold:

• (i) For every $(x,t)\in \mathbb {R}^N\times (0,\infty )$,

  $$\begin{aligned} \frac{\partial _t\Phi _{\lambda ,a}(x,t)-m(t)\Delta \Phi _{\lambda ,a}(x,t)}{\Phi _{\lambda ,a}(x,t)} \ge \frac{\eta _\lambda m(t)}{\Psi _a(x,t)}. \end{aligned}$$

• (ii) For every $(x,t)\in \mathbb {R}^N\times (0,\infty )$,

  $$\begin{aligned} \frac{c_\lambda }{\Psi _a(x,t)^{\beta _\lambda }} \le \Phi _{\lambda ,a}(x,t) \le \frac{C_\lambda }{\Psi _a(x,t)^{\beta _\lambda }}. \end{aligned}$$

• (iii) For every $(x,t)\in \mathbb {R}^N\times (0,\infty )$,

  $$\begin{aligned} \frac{|\partial _t\Phi _{\lambda ,a}(x,t)|}{\Phi _{\lambda ,a}(x,t)} \le \frac{{\widetilde{C}}_\lambda m(t)}{ \Psi _a(x,t)}, \end{aligned}$$

where the constants $\eta _\lambda$, $c_\lambda$, $C_\lambda$ and ${\widetilde{C}}_\lambda$ are given in Lemma 2.1.

Remark 3.3

The assumption $b\in {\mathcal {D}}_{\textrm{diff}}={\mathcal {D}}_*{\setminus } {\mathcal {D}}_{\textrm{over}}$ provides that $M(a;t)\rightarrow \infty$ as $t\rightarrow \infty$ and therefore the solution of $\partial _t\Phi =m(t)\Delta \Phi$ decays in some sense.

3.3 Weighted energy method for time-dependent damping

Now we derive the weight energy estimate for the solution w of (3.1). We will use the following weighted energy in the proof:

$$\begin{aligned} E_D^\lambda (w;t)&= \int _D \Big (\big (\partial _tw(x,t)\big )^2+|\nabla w(x,t)|^2\Big )\Psi _a(x,t)^\lambda \,dx, \end{aligned}$$

(3.7)

where $\Psi _a$ is given in Lemma 3.4. To state the precise estimate for w, we put

$$\begin{aligned} X_{D,a}^{\lambda }(w;t)&=M(a;t)E_D^\lambda (w;t) +\int _Dw^2\Psi _a^{\lambda }\,dx, \end{aligned}$$

(3.8)

$$\begin{aligned} Y_{D,a}^\lambda (w;t)&= \frac{M(a;t)}{m(t)}\int _D(\partial _tw)^2\Psi _a^{\lambda }\,dx + m(t)\int _D\left( |\nabla w|^2+\frac{w^2}{\Psi _a}\right) \Psi _a^{\lambda }\,dx. \end{aligned}$$

(3.9)

The following is the harvest of the above strategy.

Proposition 3.5

Assume (3.2) and $b\in {\mathcal {D}}_*$. Then there exist positive constants $C_1$, $C_2$, $C_3$ and $a>0$ (independent of R and T) such that for every $t\in [0,T]$,

$$\begin{aligned}&X_{D,a}^{\lambda }(w;t)+\int _0^t Y_{D,a}^{\lambda }(w;s)\,ds \le C_1X_{D,a}^{\lambda }(w;0)\\&\quad + C_2\int _0^t G_1(s)\,ds +C_3\int _0^tG_2(s)\,ds, \end{aligned}$$

where

$$\begin{aligned} G_1(t)=m(t)\int _D |wg|\Psi _{a}^{\lambda }\,dx, \quad G_2(t)=M(a;t)\int _D(\partial _tw)g\Psi _a^\lambda \,dx. \end{aligned}$$

Proof

As explained in the beginning of Sect. 3, we introduce the auxiliary functional

$$\begin{aligned} E_{D,a*}^\lambda (w;t) = \int _D \frac{2m(t)w(x,t)\partial _tw(x,t) +w(x,t)^2}{\Phi _{\lambda ,a}(x,t)^{1-2\delta _\lambda }} \,dx, \end{aligned}$$

(3.10)

where $\delta _\lambda =\frac{\gamma _\lambda -\beta _\lambda }{2\gamma _\lambda }=\frac{1}{2}\left( \frac{N-2\lambda }{N+2\lambda }\right) ^2\in \left[ 0,\frac{1}{2}\right)$ (which provides $(1-2\delta _\lambda )\beta _\lambda =\lambda$). We notice that since for every $a\ge a_\varepsilon$ (given in Lemma 3.2), one has

$$\begin{aligned} \frac{m(t)w\partial _tw+w^2}{\Phi _{\lambda ,a}^{1-2\delta _\lambda }}&\ge \frac{|2m(t)\partial _tw+w|^2}{4\Phi _{\lambda ,a}^{1-2\delta _\lambda }} + \frac{3w^2}{4\Phi _{\lambda ,a}^{1-2\delta _\lambda }} -\frac{m(t)^2}{c_\lambda ^{1-2\delta _\lambda }}(\partial _tw)^2\Psi _{a}^\lambda \\&\ge \frac{3w^2}{4\Phi _{\lambda ,a}^{1-2\delta _\lambda }} -\frac{\varepsilon }{c_\lambda ^{1-2\delta _\lambda }} M(a;t)(\partial _tw)^2\Psi _{a}^\lambda , \end{aligned}$$

the equivalence of the quantities

$$\begin{aligned} X_{D,a}^\lambda (w;t)\approx \nu M(a;t)E_{D,a}^\lambda (w;t) + E_{D,a*}^\lambda (w;t) \end{aligned}$$

is verified for every $\nu >0$ by taking a sufficiently large. In the final step, we will fix the parameter a. To shorten the notation, we use $\beta =\beta _\lambda$, $\gamma =\gamma _\lambda$ and $\delta =\delta _\lambda$ (without subscripts).

Let $a\ge 1$ determined later. Using the equation in the alternative form (3.4), we have

$$\begin{aligned} \frac{d}{dt} E_{D,a*}^\lambda (w;t) =J_1+J_2, \end{aligned}$$

where

$$\begin{aligned} J_1&= 2m(t)\int _D \frac{w(\Delta w +g)}{\Phi _{\lambda ,a}^{2-2\delta }}\,dx -(1-2\delta )\int _D \frac{w^2\partial _t\Phi _{\lambda ,a}}{\Phi _{\lambda ,a}^{2-2\delta }}\,dx, \\ J_2&= 2m(t)\int _D \frac{(\partial _tw)^2}{\Phi _{\lambda ,a}^{1-2\delta }}\,dx -2(1-2\delta )m(t)\int _D \frac{w\partial _tw\partial _t\Phi _{\lambda ,a}}{\Phi _{\lambda ,a}^{1-2\delta }}\,dx. \end{aligned}$$

We see from Lemmas 2.2 and 3.4(i) that

$$\begin{aligned} J_1&\le -\frac{2\delta }{1-\delta }m(t) \int _D \frac{|\nabla w|^2}{\Phi _{\lambda ,a}^{1-2\delta }}\,dx -(1-2\delta )\int _D \frac{w^2(\partial _t\Phi _{\lambda ,a}-m(t)\Delta \Phi _{\lambda ,a})}{\Phi _{\lambda ,a}^{2-2\delta }}\,dx \\&\quad +\,2m(t)\int _D \frac{w g}{\Phi _{\lambda ,a}^{1-2\delta }}\,dx \\&\le 2m(t) \left[ -\frac{\delta }{1-\delta } \int _D \frac{|\nabla w|^2}{\Phi _{\lambda ,a}^{1-2\delta }}\,dx -\frac{(1-2\delta )\eta _\lambda }{2} \int _D \frac{w^2}{\Phi _{\lambda ,a}^{1-2\delta }\Psi _a}\,dx\right. \\&\quad \left. +\,\int _D \frac{w g}{\Phi _{\lambda ,a}^{1-2\delta }}\,dx \right] . \end{aligned}$$

On the one hand, Lemma 3.4(iii) and the Young inequality yield

$$\begin{aligned} J_2&\le 2m(t) \left[ \int _D \frac{(\partial _tw)^2}{\Phi _{\lambda ,a}^{1-2\delta }}\,dx +(1-2\delta ){\widetilde{C}}_{\lambda } \int _D \frac{|w\partial _tw|}{\Phi _{\lambda ,a}^{1-2\delta }\Psi _a}\,dx \right] \\&\le 2m(t)\left[ \int _D \frac{(\partial _tw)^2}{\Phi _{\lambda ,a}^{1-2\delta }}\,dx + \frac{(1-2\delta ){\widetilde{C}}_\lambda ^2}{\eta _\lambda }\int _D \frac{(\partial _tw)^2}{\Phi _{\lambda ,a}^{1-2\delta }\Psi _a}\,dx \right. \\&\quad \left. +\,\frac{(1-2\delta )\eta _\lambda }{4} \int _D \frac{w^2}{\Phi _{\lambda ,a}^{1-2\delta }\Psi _a}\,dx \right] \\&\le 2m(t)\left[ \left( 1+\frac{(1-2\delta ){\widetilde{C}}_\lambda ^2}{\eta _\lambda }\right) \int _D \frac{(\partial _tw)^2}{\Phi _{\lambda ,a}^{1-2\delta }}\,dx\right. \\&\quad \left. +\, \frac{(1-2\delta )\eta _\lambda }{4} \int _D \frac{w^2}{\Phi _{\lambda ,a}^{1-2\delta }\Psi _a}\,dx \right] . \end{aligned}$$

Combining the above inequalities with Lemma 3.4(ii), we deduce

$$\begin{aligned}&\frac{d}{dt}E_{D,a*}^\lambda (w;t) + \frac{2\delta }{(1-\delta )C_{\lambda }^{1-2\delta }} m(t) \int _D |\nabla w|^2\Psi _a^\lambda \,dx \nonumber \\&\qquad +\, \frac{\eta _\lambda (1-2\delta )}{2 C_{\lambda }^{1-2\delta }} m(t)\int _D w^2\Psi _a^{\lambda -1}\,dx \nonumber \\&\quad \le \frac{2}{c_{\lambda }^{1-2\delta }}\left( 1+\frac{(1-2\delta ){\widetilde{C}}_\lambda ^2}{\eta _\lambda }\right) m(t)\int _D (\partial _tw)^2\Psi _a^\lambda \,dx +2m(t)\int _D \frac{w g}{\Phi _{\lambda ,a}^{1-2\delta }}\,dx. \end{aligned}$$

(3.11)

The estimate for $E_{D}^\lambda (w;t)$ is derived as follows. Using integration by parts and the equation in (3.1), we see that

$$\begin{aligned} \frac{d}{dt} \Big [M(a;t)E_{D,a}^\lambda (w;t)\Big ] =J_3+J_4+J_5, \end{aligned}$$

where

$$\begin{aligned} J_3&= 2M(a;t)\int _D \partial _tw(-b(t)\partial _tw+g)\Psi _a^{\lambda }\,dx, \\ J_4&= 2\lambda M(a;t)\int _D \partial _tw\nabla w\cdot \nabla \Psi _a \Psi _a^{\lambda -1}\,dx, \\ J_5&= \int _D \Big ((\partial _tw)^2+|\nabla w|^2\Big )\partial _t[M(a;t)\Psi _a^\lambda ]\,dx. \end{aligned}$$

Then Lemma 3.2 gives

$$\begin{aligned} J_3&\le -2B_0\frac{M(a;t)}{m(t)}\int _D (\partial _tw)^2\Psi _a^{\lambda }\,dx + 2G_2(t). \end{aligned}$$

The relation $M(a;t)+|\nabla \Psi _a|^2=\Psi _a$ with the Young inequality provides

$$\begin{aligned} J_4&\le 2\lambda \sqrt{M(a;t)}\int _D |\partial _tw||\nabla w|\Psi _a^{\lambda }\,dx \\&\le B_0\frac{M(a;t)}{m(t)}\int _D (\partial _tw)^2\Psi _a^{\lambda }\,dx + \frac{\lambda ^2}{B_0} m(t)\int _D |\nabla w|^2\Psi _a^{\lambda }\,dx \end{aligned}$$

and

$$\begin{aligned} J_5\le (\lambda +1)m(t)E_{D}^\lambda (w;t). \end{aligned}$$

Summarizing the above estimates, we obtain

$$\begin{aligned}&\frac{d}{dt} \Big [M(a;t)E_{D,a}^{\lambda }(w;t)\Big ] + B_0 \frac{M(a;t)}{m(t)} \int _D (\partial _tu)^2\Psi _a^{\lambda }\,dx \nonumber \\&\quad \le (\lambda +1) m(t)\int _D (\partial _tu)^2\Psi _a^{\lambda }\,dx + \Big (\lambda +1+\frac{\lambda ^2}{B_0}\Big ) m(t)\int _D |\nabla u|^2\Psi _a^{\lambda }\,dx +2G_2(t). \end{aligned}$$

(3.12)

Choosing

$$\begin{aligned} \nu = \frac{\delta }{(1-\delta )C_\lambda ^{1-2\delta }} \Big (\lambda +1+\frac{\lambda ^2}{B_0}\Big )^{-1}, \end{aligned}$$

by (3.11) and (3.12) we can deduce

$$\begin{aligned}&\frac{d}{dt} \Big [\nu M(a;t)E_{D,a}^{\lambda }(w;t)+E_{D,a*}^\lambda (w;t)\Big ] + \nu B_0 \frac{M(a;t)}{m(t)} \int _D (\partial _tu)^2\Psi _a^{\lambda }\,dx \\&\qquad + \,\frac{\delta }{(1-\delta )C_{\lambda }^{1-2\delta }} m(t) \int _D |\nabla w|^2\Psi _a^\lambda \,dx + \frac{\eta _\lambda (1-2\delta )}{2 C_{\lambda }^{1-2\delta }} \int _D w^2\Psi _a^{\lambda -1}\,dx \\&\quad \le \left[ \nu (\lambda +1) + \frac{2(1+{\widetilde{C}}_\lambda ^2)}{c_\lambda ^{1-2\delta }} \right] m(t)\int _D (\partial _tu)^2\Psi _a^{\lambda }\,dx +\frac{2}{c_\lambda ^{1-2\delta }}G_1(t)+2\nu G_2(t). \end{aligned}$$

Here choosing a suitable parameter $a\ge 1$ such that the two quantities $X_{D,a}^\lambda (w;t)$ and $\nu M(a;t)E_{D,a}^{\lambda }(w;t)+E_{D,a*}^\lambda (w;t)$ are equivalent and

$$\begin{aligned} \frac{m(t)^2}{M(a;t)} \le \frac{\nu B_0}{2} \left[ \nu (\lambda +1) + \frac{2(1+{\widetilde{C}}_\lambda ^2)}{c_\lambda ^{1-2\delta }} \right] ^{-1} \end{aligned}$$

(by virtue of Lemma 3.3), we obtain the desired inequality. $\square$

Proof of Theorem 1.2

Set

$$\begin{aligned} R_\Omega = {\left\{ \begin{array}{ll} \sup \{|x|;\;x\in \Omega ^c\}&{}\text {if}\ \Omega \ne \mathbb {R}^N,\\ 0&{}\text {if}\ \Omega = \mathbb {R}^N. \end{array}\right. } \end{aligned}$$

Let $T>0$ be arbitrary. If ${{\,\textrm{supp}\,}}u_0 \cup {{\,\textrm{supp}\,}}u_1$ is compact in ${\overline{\Omega }}$, then we fix a constant $R>R_\Omega$ satisfying ${{\,\textrm{supp}\,}}u_0 \cup {{\,\textrm{supp}\,}}u_1\subset B(0,R)$. By finite speed of propagation, we have ${{\,\textrm{supp}\,}}u(\cdot ,t)\subset B(0,R+t)$ for every $t\in [0,T]$. This means that u also satisfies (3.1) with $g\equiv 0$ and $D=\Omega \cap B(0,R+T)$. Applying Proposition 3.5, we arrive the weighted energy estimate

$$\begin{aligned} \int _\Omega \Big ((\partial _tu)^2+|\nabla u|^2\Big )W^\lambda \,dx = \int _D\Big ((\partial _tu)^2+|\nabla u|^2\Big )W^\lambda \,dx \le C I_{\Omega }^\lambda (u_0,u_1), \quad t\in [0,T] \end{aligned}$$

with the constant C which is independent of R and also T. Since T is arbitrary, we obtain the desired estimate.

If ${{\,\textrm{supp}\,}}u_0 \cup {{\,\textrm{supp}\,}}u_1$ is not compact, then the standard cut-off argument with the strong continuity of the $C_0$-semigroup $e^{t{\mathcal {L}}}$ in $H_0^1(\Omega )\times L^2(\Omega )$ provides the desired estimate. $\square$

4 Nonlinear estimates for global existence

Finally, we discuss the existence of global-in-time weak solutions to (1.1). As in the linear case, we additionally assume that ${{\,\textrm{supp}\,}}u_0 \cup {{\,\textrm{supp}\,}}u_1\subset B(0,R)$ for some $R>R_\Omega$. Let u be the solution of (1.1) in (0, T) given in Proposition 1.1. Then by finite speed of propagation we also have ${{\,\textrm{supp}\,}}u(\cdot ,t)\subset B(0,R+t)$ for every $t\in [0,T)$. As in the proof of Theorem 1.2, we consider the homogeneous Dirichlet boundary problem in bounded domain $D=\Omega \cap B(0,R+T)$ and use the functionals

$$\begin{aligned} X(t)=X_{D,a}^\lambda (u;t) \quad \text {and}\quad Y(t)=Y_{D,a}^\lambda (u;t) \end{aligned}$$

(for the simplicity of the notations) introduced in (3.8) and (3.9), respectively. Then we define a non-decreasing continuous function

$$\begin{aligned} Z(t)=\sup _{0\le s\le t}X(s)+\int _0^tY(s)\,ds, \quad t\in (0,T); \end{aligned}$$

note that in the linear case ($f\equiv 0$), by Proposition 3.5, Z(t) $(t>0)$ is bounded.

The following proposition is crucial to obtain the uniform bound for Z in t.

Proposition 4.1

Assume $b\in {\mathcal {D}}_{\textrm{diff}}$ and (1.2) with $p(N,\lambda )\le p<p_*(N)$ for some $\lambda \in [0,\frac{N}{2})$. Assume further that ${{\,\textrm{supp}\,}}u_0 \cup {{\,\textrm{supp}\,}}u_1$ is compact in ${\overline{\Omega }}$. Let u be the local-in-time solution of (1.1) in (0, T). Then there exists a positive constant C (independent of R and T) such that for every $t\in (0,T)$,

$$\begin{aligned} Z(t) \le C \left( X(0)+X(0)^{\frac{p+1}{2}}+Z(t)^{\frac{p+1}{2}}\right) . \end{aligned}$$

The following lemma is the central part of the proof of Proposition 4.1 which treats the following quantity associated to the nonlinear term f(u);

$$\begin{aligned} {\widetilde{F}}(t)=\int _D|u(x,t)|^{p+1}\Psi _a(x,t)^\lambda \,dx, \quad t\in [0,T). \end{aligned}$$

(4.1)

Lemma 4.2

Let ${\widetilde{F}}(t)$ as in (4.1). Under the assumption in Proposition 4.1, one has for every $t\in (0,T)$,

$$\begin{aligned} M(a;t) {\widetilde{F}}(t)\le & {} K_{\lambda ,p}a^{1-\frac{N+2\lambda }{4}(p-1)}\Vert X\Vert _{L^{\infty }(0,t)}^{\frac{p+1}{2}}, \\ \int _0^tm(s) {\widetilde{F}}(s) \,ds\le & {} K_{\lambda ,p}a^{1-\frac{N+2\lambda }{4}(p-1)}\Vert X\Vert _{L^{\infty }(0,t)}^{\frac{p-1}{2}} \Vert Y\Vert _{L^{1}(0,t)}. \end{aligned}$$

Proof

Here we put $q=1+\frac{4}{N}$ and divide the proof into two cases where $p \ge q$ and $p(N,\lambda )\le p <q$.

(The case $p \ge q=1+\frac{4}{N}$) Adopting the Gagliardo–Nirenberg inequality (Lemma 2.3 with $\tau =M(a;t)$, $\mu =\lambda$), we deduce

$$\begin{aligned} {\widetilde{F}}(t)\!\le \! \frac{K_{\lambda ,p}}{(M(a;t))^{\frac{\lambda }{2}(p\!-\!1)}} \left( \int _Du^2\Psi _a^\lambda \,dx\right) ^{\frac{p+1}{2}\!-\!\frac{N}{4}(p\!-\!1)} \left( \int _D\left( |\nabla u|^2+\frac{u^2}{\Psi _a}\right) \Psi _a^\lambda \,dx\right) ^{\frac{N}{4}(p-1)}. \end{aligned}$$

Therefore the relation $\frac{N}{4}(p-1)\ge 1$ implies the following two kind of inequalities

$$\begin{aligned} M(a;t) {\widetilde{F}}(t)&\le K_{\lambda ,p}\big (M(a;t)\big )^{1-\frac{N+2\lambda }{4}(p-1)}X(t)^{\frac{p+1}{2}}, \\ m(t){\widetilde{F}}(t)&\le K_{\lambda ,p}\big (M(a;t)\big )^{-\frac{N+2\lambda }{4}(p-1)} X(t)^{\frac{p-1}{2}}Y(t). \end{aligned}$$

(The case $p(N,\lambda )\le p <q$) The first inequality in the assertion can be verified the computation exactly the same as above. For the second inequality, the Hölder inequality and the Gagliardo–Nirenberg inequality yield

$$\begin{aligned} \int _D|u|^{p+1}\Psi _a^\lambda \,dx \le \left( \int _D \frac{u^2}{\Psi _a}\Psi _a^\lambda \,dx \right) ^{1-\frac{N}{4}(p-1)} \left( \int _D|u|^{q+1}\Psi _a^{\frac{\lambda }{2}(q+1)}\,dx \right) ^{\frac{N}{4}(p-1)} \end{aligned}$$

and

$$\begin{aligned} \int _D|u|^{q+1}\Psi _a^{\frac{\lambda }{2}(q+1)}\,dx&\le K_{0,p} \left( \int _D u^2\Psi _a^\lambda \,dx \right) ^{\frac{2}{N}} \int _D|\nabla (u\Psi _a^{\frac{\lambda }{2}})|^2\,dx. \\&\le \!2\max \{1,\lambda \}K_{0,p} \left( \int _D u^2\Psi _a^\lambda \,\!dx \right) ^{\frac{2}{N}}\! \int _D\left( |\nabla u|^2\!+\!\frac{u^2}{\Psi _a}\right) \!\Psi _a^\lambda \,dx. \end{aligned}$$

These inequalities give the desired (second) inequality. $\square$

Proof of Proposition 4.1

Set $F(\xi )=\int _0^\xi f(\theta )\,d\theta$ ($\xi \in \mathbb {R}$) and the corresponding external functions in Proposition 3.5 as

$$\begin{aligned} {\widetilde{G}}_1(t)=m(t)\int _D |uf(u)|\Psi _{a}^{\lambda }\,dx, \quad {\widetilde{G}}_2(t)=M(a;t)\int _D\partial _tu f(u)\Psi _a^\lambda \,dx, \end{aligned}$$

respectively. Then we see from (1.2) that

$$\begin{aligned} \int _0^t {\widetilde{G}}_1(s)\,ds \le C_f \int _0^t m(s) {\widetilde{F}}(s) \,ds. \end{aligned}$$

On the other hand, noting that $|F(\xi )|\le \frac{C_f}{p+1}|\xi |^{p+1}$ and

$$\begin{aligned} {\widetilde{G}}_2(t) = \frac{d}{dt}\Big [M(a;t)\int _DF(u)\Psi _a^\lambda \,dx\Big ] - \int _DF(u)\partial _t\big [M(a;t)\Psi _a^\lambda \big ]\,dx, \end{aligned}$$

we have

$$\begin{aligned} \int _0^t {\widetilde{G}}_2(s)\,ds&\le \frac{C_f}{p+1}M(a;t){\widetilde{F}}(s) + \frac{C_f}{p+1}M(a;0){\widetilde{F}}(0)\\&\quad + \frac{(\lambda +1)C_f}{p+1} \int _0^tm(s){\widetilde{F}}(s) \,ds. \end{aligned}$$

Therefore combining Proposition 3.5 and the above inequalities together with Lemma 4.2, we obtain the desired inequality. $\square$

Proof of Theorem 1.3

For the case of initial data $(u_0,u_1)\in {\mathcal {H}}$ having compact supports, then we see from Proposition 4.1 that

$$\begin{aligned} {\widetilde{Z}}(t) \le {\widetilde{X}}(0)+{\widetilde{Z}}(t)^{\frac{p+1}{2}}, \quad t\in [0,T). \end{aligned}$$

where we have put ${\widetilde{X}}(0)=C^{\frac{p+1}{p-1}}(X(0)+X(0)^{\frac{p+1}{2}})$ and ${\widetilde{Z}}(t)=C^{\frac{2}{p-1}}Z(t)$. By noticing the (non-)connectedness of $\{z\ge 0;\Theta _\delta (z)=\delta +z^{\frac{p+1}{2}}-z\ge 0\}$ with the convexity of $\Theta _\delta$, we can check that if

$$\begin{aligned} {\widetilde{X}}(0)<\frac{p-1}{2}\left( \frac{2}{p+1}\right) ^{\frac{p+1}{p-1}}, \end{aligned}$$

then one has

$$\begin{aligned} {\widetilde{Z}}(t)\le \frac{p+1}{p-1}{\widetilde{X}}(0) \end{aligned}$$

which is nothing but the desired uniform estimate of the weighted energy

$$\begin{aligned} (1+B(t))\int _\Omega \Big (|\nabla u|^2+(\partial _tu)^2\Big )W^{\lambda }\,dx + \int _\Omega u^2W^\lambda \,dx \le {\widetilde{C}} {\widetilde{X}}(0). \end{aligned}$$

By the argument with the blowup criteria, we can construct a global-in-time weak solution of (1.1) satisfying the above uniform estimate.

If we consider the case of initial data with non-compact supports, then the approximation procedure for the initial data via a family of cut-off functions with the previous step provides approximate (global-in-time weak) solutions $u_n$ with the uniform weighted energy estimate. Then we can show that the limit $\lim _{n\rightarrow \infty }u$ (taking a subsequence if necessary) is the global-in-time solution of (1.1) with the given initial data $(u_0,u_1)$. $\square$

Remark 4.1

If $b\in {\mathcal {D}}_{*}\cap {\mathcal {D}}_{\textrm{over}}$, then M(a; t) is bounded in t and therefore it is enough to choose $\lambda =0$ and then for every $p>1$,

$$\begin{aligned} M(a;t)\Vert u(t)\Vert _{L^{p+1}(D)}^{p+1}\le & {} K_{0,p} \left( \int _Du^2\,dx\right) ^{\frac{p+1}{2}-\frac{N}{4}(p-1)} \left( \int _D|\nabla u|^2\,dx\right) ^{\frac{N}{4}(p-1)} \\\le & {} C_pX(t)^{\frac{p+1}{2}} \end{aligned}$$

holds for some constant $C_p$ (independent of T and also R). We also have

$$\begin{aligned} \int _0^t m(s) M(a;t)\Vert u(s)\Vert _{L^{p+1}(D)}^{p+1} \le C_pX(t)^{\frac{p+1}{2}}\int _0^tm(s)\,ds \le {\widetilde{C}}_pX(t)^{\frac{p+1}{2}}. \end{aligned}$$

This means that the strategy of the proof of Theorem 1.3 also works when $1<p<p_*(N)$ and b belongs to a class of overdamping coefficients.