Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg ( IF 0.4 ) Pub Date : 2022-12-13 , DOI: 10.1007/s12188-022-00262-6 M. Abdellaoui , H. Redwane
We study the existence and uniqueness of renormalized solutions for initial boundary value problems of the type
$$\begin{aligned} \left( {\mathcal {P}}_{b}^{1}\right) \quad \left\{ \begin{aligned} u_{t}-\text {div}(a(t,x,\nabla u))=H(u)\mu \text { in }Q:=(0,T)\times \Omega ,\\ u(0,x)=u_{0}(x)\text { in }\Omega ,\ u(t,x)=0\text { on }(0,T)\times \partial \Omega , \end{aligned}\right. \end{aligned}$$where \(u_{0}\in L^{1}(\Omega )\), \(\mu \in {\mathcal {M}}_{b}(Q)\) is a general Radon measure on Q and \(H\in C_{b}^{0}({\mathbb {R}})\) is a continuous positive bounded function on \({\mathbb {R}}\). The difficulties in the study of such problems concern the possibly very singular right-hand side that forces the choice of a suitable formulation that ensures both existence and uniqueness of solution. Using similar techniques, we will prove existence/nonexistence results of the auxiliary problem
$$\begin{aligned} \left( {\mathcal {P}}_{b}^{2}\right) \quad \left\{ \begin{aligned}&u_{t}-\text {div}(a(t,x,\nabla u))+g(x,u)|\nabla u|^{2}=\mu \text { in }Q:=(0,T)\times \Omega ,\\&u(0,x)=u_{0}(x)\text { in }\Omega ,\ u(t,x)=0\text { on }(0,T)\times \partial \Omega , \end{aligned}\right. \end{aligned}$$under the assumption that g satisfies a sign condition and the nonlinear term depends on both x, u and its gradient. Thus, our results improve and complete the previous known existence results for problems \(\left( {\mathcal {P}}_{b}^{1,2}\right) \).
中文翻译：
右手边可能非常奇异的初边值抛物线问题重整化解的存在唯一性
我们研究了该类型初始边值问题重整化解的存在唯一性
$$\begin{aligned} \left( {\mathcal {P}}_{b}^{1}\right) \quad \left\{ \begin{aligned} u_{t}-\text {div}( a(t,x,\nabla u))=H(u)\mu \text { }Q:=(0,T)\times \Omega ,\\ u(0,x)=u_{0}( x)\text { in }\Omega ,\ u(t,x)=0\text { on }(0,T)\times \partial \Omega , \end{aligned}\right. \end{对齐}$$其中\(u_{0}\in L^{1}(\Omega )\) , \(\mu \in {\mathcal {M}}_{b}(Q)\)是Q上的一般氡测量并且\(H\in C_{b}^{0}({\mathbb {R}})\)是\({\mathbb {R}}\)上的连续正有界函数。研究此类问题的困难在于右侧可能非常奇异，迫使选择合适的公式来确保解的存在性和唯一性。使用类似的技术，我们将证明辅助问题的存在/不存在结果
$$\begin{aligned} \left( {\mathcal {P}}_{b}^{2}\right) \quad \left\{ \begin{aligned}&u_{t}-\text {div}( a(t,x,\nabla u))+g(x,u)|\nabla u|^{2}=\mu \text { 在 }Q:=(0,T)\times \Omega ,\\ &u(0,x)=u_{0}(x)\text { in }\Omega ,\ u(t,x)=0\text { on }(0,T)\times \partial \Omega , \end {对齐}\右。\end{对齐}$$假设g满足符号条件并且非线性项取决于x、u及其梯度。因此，我们的结果改进并完善了先前已知的问题存在性结果\(\left( {\mathcal {P}}_{b}^{1,2}\right) \)。
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