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Existence and uniqueness of renormalized solutions for initial boundary value parabolic problems with possibly very singular right-hand side

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Abstract

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 \(\left( {\mathcal {P}}_{b}^{1,2}\right) \).

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Notes

  1. \(\psi : L^{2}(0,T;H^{1}_{0}(\Omega ))\rightarrow L^{2}(0,T;H^{1}_{0}(\Omega ))\cap L^{\infty }(Q)\) is an increasing function.

  2. Observe that \(\psi \) is a \(C^{1}\)-function satisfying

    $$\begin{aligned} \left\{ \begin{aligned}&(p-1)\psi ''(s)=\psi '(s)^{2}-\frac{\psi '(s)g(s)}{\beta },\\&\psi (0)=0\text { and }\psi '(0)=\frac{p-1}{S}. \end{aligned}\right. \end{aligned}$$

  3. By a little abuse of notation, we write \(T_{L}(u)\) for the \(\text {cap}_{2}\)-quasi continuous represnetative of \(T_{L}(u)\) which is uniquely defined since the truncations of u lies in \(L^{2}(0,T;H^{1}_{0}(\Omega ))\).

  4. A weak solution of problem (3.1) is also a renormalized solution of the same problem.

  5. Such an approximation exists, it can be obtained by using a partition of unity, and then locally approximating by means of a suitably decentered convolution Kernel.

  6. It can also be obtained by using Schauder’s fixed point theorem.

  7. Observe that \(\int _{0}^{T_{2n}(u_{n})}|\psi '(s)|ds\le \Vert \psi '\Vert _{L^{1}({\mathbb {R}}^{+})}\).

  8. Observe that \(\psi \) is bounded since \(\psi '\) belongs to \(L^{1}({\mathbb {R}}^{+})\).

  9. Observe that \((k-u_{n})^{+}=0\) on the subset \(\lbrace (t,x)\in Q:u_{n}(t,x)>k\rbrace \) and that \(h_{n}(u_{n})=1\) for \(n>k\).

  10. \(\text {supp }\theta _{n}(s)=\lbrace |s|\ge n\rbrace \).

  11. The existence of the constants \(k>0\) and \(C_{k}>0\) comes from the fact that \(S'\) has compact support.

  12. Observe that two different solutions u and v of the same problem satisfy

    $$\begin{aligned} \int _{Q}(u-v)\psi dxdt,\quad \forall \psi \in C^{\infty }_{0}(Q). \end{aligned}$$

  13. Remark that \(\theta '_{n}\) has compact support and \(\theta _{n}(\infty )=1\).

  14. Observe that \(1-\theta _{n}(u_{n})\) is not zero only on the set where \(\lbrace (t,x)\in Q:|u_{n}(t,x)|\le 2n\rbrace \).

References

  1. Abdellaoui, M.: Asymptotic behavior of solutions for nonlinear parabolic operators with natural growth term and measure data. J. Pseudo Differ. Oper. Appl. 1–41 (2019)

  2. Abdellaoui, M., Azroul, E.: Renormalized solutions for nonlinear parabolic equations with general measure data. Electron. J. Differ. Equ. 132, 1–21 (2018)

    MathSciNet  MATH  Google Scholar 

  3. Abdellaoui, M., Redwane, H.: On some regularity results of parabolic problems with nonlinear perturbed terms and general data. Partial Differ. Equ. Appl. 3, 1–39 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  4. Adams, D.R., Hedberg, L.I.: Function Spaces and Potential Theory, Grundlehren der mathematischen Wissenschaften, 314. Springer-Verlag, Berlin (1996)

    Google Scholar 

  5. Boccardo, L., Dall’Aglio, A., Gallouët, T., Orsina, L.: Nonlinear parabolic equations with measure data. J. Funct. Anal. 147, 237–258 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  6. Boccardo, L., Gallouët, T., Orsina, L.: Existence and uniqueness of entropy solutions for nonlinear elliptic equations with measure data. Ann. Inst. H. Poincaré Anal. Non Linéaire 13, 539–551 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  7. Boccardo, L., Gallouët, T., Orsina, L.: Existence and nonexistence of solutions for some nonlinear elliptic equations. J. Anal. Math. 73, 203–223 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  8. Bénilan, P., Boccardo, L., Gallouët, T., Gariepy, R., Pierre, M., Vázquez, J.L.: An \(L^{1}\)-theory of existence and uniqueness of nonlinear elliptic equations. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 22, 241–273 (1995)

    MathSciNet  MATH  Google Scholar 

  9. Blanchard, D., Porretta, A.: Nonlinear parabolic equations with natural growth terms and measure initial data. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 4(30), 583–622 (2001)

    MathSciNet  MATH  Google Scholar 

  10. Boccardo, L., Murat, F.: Almost everywhere convergence of the gradients of solutions to elliptic and parabolic equations. Nonlinear Anal. 19, 581–597 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  11. Boccardo, L., Murat, F., Puel, J.P.: Existence of bounded solutions for nonlinear elliptic unilateral problems. Ann. Mat. Pura Appl. 152, 183–196 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  12. Biegert, M.: Elliptic problems on varying domains. Dissertation, Logos Verlag, Berlin (2005)

  13. Boccardo, L., Gallouët, T.: Nonlinear elliptic and parabolic equations involving measure data. J. Funct. Anal. 87, 149–169 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  14. Boccardo, L., Gallouët, T.: Nonlinear elliptic equations with right hand side measures. Commun. Partial Differ. Equ. 17(3 &4), 641–655 (1992)

    MathSciNet  MATH  Google Scholar 

  15. Brezis, H., Nirenberg, L.: Removable singularities for nonlinear elliptic equations. Topol. Methods Nonlinear Anal. 9, 201–219 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  16. Choquet, G.: Theory of capacities. Ann. Inst. Fourier 5, 131–295 (1954)

    Article  MathSciNet  MATH  Google Scholar 

  17. Dal Maso, G., Murat, F., Orsina, L., Prignet, A.: Definition and existence of renormalized solutions of elliptic equations with general measure data. Comptes Rendus Math. Acad. Sci. Paris Ser. I(325), 481–486 (1997)

    MathSciNet  MATH  Google Scholar 

  18. Dal Maso, G., Murat, F., Orsina, L., Prignet, A.: Renormalized solutions of elliptic equations with general measure data. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 28, 741–808 (1999)

    MathSciNet  MATH  Google Scholar 

  19. Dall’Aglio, A., Orsina, L.: Existence results for some nonlinear parabolic equations with nonregular data. Differ. Integral Equ. 5, 1335–1354 (1992)

    MathSciNet  MATH  Google Scholar 

  20. De Cave, L.M., Oliva, F.: Elliptic equations with general singular lower order term and measure data. Nonlinear Anal. 128, 391–411 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  21. Dall’Aglio, A., Orsina, L., Petitta, F.: Existence of solutions for degenerate parabolic equations with singular terms. Nonlinear Anal. 131, 273–288 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  22. De Cave, L.M., Durastanti, R., Oliva, F.: Existence and uniqueness results for possibly singular nonlinear elliptic equations with measure data. NoDEA Nonlinear Differ. Equ. Appl. 25(3), 18–35 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  23. DiPerna, R.-J., Lions, P.-L.: On the Cauchy problem for Boltzmann equations, global existence and weak stability. Ann. Math. 130, 321–366 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  24. Droniou, J., Prignet, A.: Equivalence between entropy and renormalized solutions for parabolic equations with smooth measure data. NoDEA 14(1–2), 181–205 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  25. Droniou, J., Porretta, A., Prignet, A.: Parabolic capacity and soft measures for nonlinear equations. Potential Anal. 19(2), 99–161 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  26. Fukushima, M., Sato, K., Taniguchi, S.: On the closable part of pre-Dirichlet forms and the fine supports of underlying measures. Osaka J. Math. 28, 517–535 (1991)

    MathSciNet  MATH  Google Scholar 

  27. Heinonen, J., Kilpeläinen, T., Martio, O.: Nonlinear Potential Theory of Degenerate Elliptic Equations. Oxford University Press, Oxford (1993)

    MATH  Google Scholar 

  28. Landes, R.: On the existence of weak solutions for quasilinear parabolic boundary value problems. Proc. R. Soc. Edinb. Sect. A 89, 217–237 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  29. Leray, J., Lions, J.-L.: Quelques résultats de Višik sur les problèmes elliptiques semi-linéaires par les méthodes de Minty et Browder. Bull. Soc. Math. France 93, 97–107 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  30. Leonori, T., Petitta, F.: Asymptotic behavior of solutions for parabolic equations with natural growth term and irregular data. Asymptot. Anal. 48(3), 219–233 (2006)

    MathSciNet  MATH  Google Scholar 

  31. Lions, J.-L.: Quelques méthodes de résolution des problèmes aux limites non linéaire. Dunod et Gauthier-Villars, Paris (1969)

    MATH  Google Scholar 

  32. Lions, P.L., Murat, F.: Sur les solutions renormalisées d’équations elliptiques non linéaires. Preprint

  33. Maz’ya, V.G.: Sobolev Spaces. Springer, Berlin (1985)

    Book  MATH  Google Scholar 

  34. Malusa, A., Prignet, A.: Stability of renormalized solutions of elliptic equations with measure data, Atti Semin. Mat. Fis. Univ. Modena Reggio Emilia 52(2004), 151–168 (2005)

    MATH  Google Scholar 

  35. Malusa, A.: A new proof of the stability of renormalized solutions to elliptic equations with measure data. Asymptot. Anal. 43(1–2), 111–129 (2005)

    MathSciNet  MATH  Google Scholar 

  36. Malusa, A., Porzio, M.M.: Renormalized solutions to elliptic equations with measure data in unbounded domains. Nonlinear Anal. TMA 67, 2370–2389 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  37. Magliocca, M., Oliva, F.: On some parabolic equations involving superlinear singular gradient terms. J. Evol. Equ. 21(2), 2547–2590 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  38. Murat, F.: Soluciones renormalizadas de EDP elipticas no lineales, Preprint 93023. Laboratoire d’Analyse Numérique de l’Université Paris, VI (1993)

  39. Murat, F.: Équations elliptiques non linéaires avec second membre \(L^{1}\) ou mesure, In Comptes Rendus du 26ème Congrés National d’Analyse Numérique, Les Karellis, A12–A24 (1994)

  40. Murat, F., Porretta, A.: Stability properties, existence and nonexistence of renormalized solutions for elliptic equations with measure data. Commun. Partial Differ. Equ. 27, 2267–2310 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  41. Oliva, F., Petitta, F.: A nonlinear parabolic problem with singular terms and nonregular data. Nonlinear Anal. 194, 111472 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  42. Petitta, F.: Renormalized solutions of nonlinear parabolic equations with general measure data. Ann. Mat. Pura Appl. 187(4), 563–604 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  43. Petitta, F.: Asymptotic behavior of solutions for linear parabolic equations with general measure data. Comptes Rendus Acad. Sci. Paris Ser. I344, 571–576 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  44. Petitta, F.: Asymptotic behavior of solutions for parabolic operators of Leray-Lions type and measure data. Adv. Differ. Equ. 12(8), 867–891 (2007)

    MathSciNet  MATH  Google Scholar 

  45. Petitta, F.: Large time behavior for solutions of nonlinear parabolic problems with sign-changing measure data. Electron. J. Differ. Equ. 132, 1–10 (2008)

    MathSciNet  MATH  Google Scholar 

  46. Pierre, M.: Parabolic capacity and Sobolev spaces. SIAM J. Math. Anal. 14, 522–533 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  47. Porretta, A.: Elliptic and parabolic equations with natural growth terms and measure data. Ph.D. Thesis, Rome (1999)

  48. Porretta, A.: Existence results for nonlinear parabolic equations via strong convergence of truncations. Ann. Mat. Pura Appl. 177, 143–172 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  49. Porretta, A.: Some remarks on the regularity of solutions for a class of elliptic equations with measure data. Houst. J. Math. 26, 183–213 (2000)

    MathSciNet  MATH  Google Scholar 

  50. Porretta, A.: Existence for elliptic equations in \(L^{1}\) having lower order terms with natural growth. Port. Math. 57, 179–190 (2000)

    MATH  Google Scholar 

  51. Prignet, A.: Remarks on existence and uniqueness of solutions of elliptic problems with right hand side measures. Rend. Mat. 15, 321–337 (1995)

    MathSciNet  MATH  Google Scholar 

  52. Prignet, A.: Existence and uniqueness of entropy solutions of parabolic problems with \(L^{1}\) data. Nonlinear Anal. TMA 28, 1943–1954 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  53. Schwartz, L.: Théorie des distributions à valeurs vectorielles I. In: Ann. Inst. Fourier, Grenoble, vol. 7, pp. 1–141 (1957)

  54. Stampacchia, G.: Le problème de Dirichlet pour les équations elliptiques du seconde ordre à coefficientes discontinus. Ann. Inst. Fourier 15, 189–258 (1965)

    Article  MATH  Google Scholar 

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Authors and Affiliations

  1. Laboratory LAMA, Department of Mathematics, Faculty of Sciences Dhar El Mahraz, University of Fez, B.P. 1796, Atlas Fez, Morocco

    M. Abdellaoui

  2. Faculté d’économie et de gestion, Université Hassan 1, B.P. 764, Settat, Morocco

    H. Redwane

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  1. M. Abdellaoui
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  2. H. Redwane
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Abdellaoui, M., Redwane, H. Existence and uniqueness of renormalized solutions for initial boundary value parabolic problems with possibly very singular right-hand side. Abh. Math. Semin. Univ. Hambg. 92, 209–245 (2022). https://doi.org/10.1007/s12188-022-00262-6

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