О существовании вязких решений эволюционного уравнения с p(x)- лапласианом с одной пространственной переменной

О существовании вязких решений эволюционного уравнения с $p(x)$-лапласианом с одной пространственной переменной

Терсенов А. С.
Сибирский журнал индустриальной математики, 27, 4(100), 130-151 (2024)
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УДК 517.95 
DOI: 10.33048/SIBJIM.2024.27.409

Аннотация:

В настоящей статье изучается первая краевая задача для уравнения с $p(x)$-лапласианом с одной пространственной переменной при наличии градиентных членов, не удовлетворяющих условию Бернштейна—Нагумо. Определён класс градиентных нелинейностей, для которого доказано существование вязкого по Лионсу решения непрерывного по Липшицу по $x$ и по Гёльдеру по $t$.

Литература:
  1. Acerbi E., Mingione G. Regularity results for stationary electro-rheological fluids // Arch. Ration. Mech. Anal. 2002. V. 164. P. 213–259.
     
  2. Antontsev S. N., Rodrigues J. F. On stationary thermo-rheological viscous flows // Ann. Univ. Ferrara, Sez. VII Sci. Mat. 2006. V. 52, N 1. P. 19–36.
     
  3. Rajagopal K., Ru$\check{z}$i$\check{c}$ka M. Mathematical modelling of electro-rheological fluids // Contin. Mech. Thermodyn. 2001. V. 13. P. 59–78.
     
  4. Ru$\check{z}$i$\check{c}$ka M. Electrorheological Fluids: Modeling and Mathematical Theory. Berlin: Springer, 2000.
     
  5. Aboulaicha R., Meskinea D., Souissia A. Regularity results for stationary electro-rheological fluids // Arch. Ration. Mech. Anal. 2002. V. 164. P. 213–259.
     
  6. Chen Y., Levine S., Rao M. Variable exponent, linear growth functionals in image restoration // SIAM J. Appl. Math. 2006. V. 66, N 4. P. 1383–1406.
     
  7. Antontsev S., Shmarev S. Evolution PDEs with Nonstandard Growth Conditions: Existence, Uniqueness, Localization, Blow-up. Paris: Atlantis Press. 2015.
     
  8. Arora R., Shmarev S. Strong solutions of evolution equations with $p(x, t)$-Laplacian: Existence, global higher integrability of the gradients and second-order regularity // J. Math. Anal. Appl. 2020. V. 493, N 1. P. 1–31.
     
  9. Arora R., Shmarev S. Existence and regularity results for a class of parabolic problems with double phase flux of variable growth // Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. AMat. 2023. V. 117, N 34. P. 1–48.
     
  10. Belloni M., Kawohl B. The pseudo-$p$-Laplace eigenvalue problem and viscosity solutions as $p \to \infty$ // ESAIM: Control Optim. Calc. Variations. 2004. V. 10, N 1. P. 28–52.
     
  11. Birindelli I., Demengel F. Existence and regularity results for fully nonlinear operators on the model of the pseudo Pucci’s operators // J. Elliptic Parabol. Equ. 2016. V. 2. P. 171–187.
     
  12. Demengel F. Lipschitz interior regularity for the viscosity and weak solutions of the pseudo $p$-Laplacian equation // Adv. Differ. Equ. 2016. V. 21, N 3. P 373–400.
     
  13. Juutinen P. On the definition of viscosity solutions for parabolic equations // Proc. Amer. Math. Soc. 2001. V. 129, N 10. P. 2907–2911.
     
  14. Tersenov Ar. S. Viscosity subsolutions and supersolutions for non-uniformly and degenerate elliptic equations // Arch. Math. 2009. V. 45, N 1. P. 19–35.
     
  15. Juutinen P., Lindqvist P., Manfredi J. J. On the equivalence of the viscosity solutions and weak solutions for a quasilinear equation // SIAM J. Math. Anal. 2001. V. 33, N 3. P. 699–717.
     
  16. Medina M., Ochoa P. On viscosity and weak solutions for non-homogeneous $p$-Laplace equations // Adv. Nonlinear Anal. 2019. V. 8, N 1. P. 468–481.
     
  17. Siltakoski J. Equivalence of viscosity and weak solutions for a $p$-parabolic equation // J. Evol. Equ. 2021. V. 21, N 4. P. 2047–2080.
     
  18. Dall’Aglio A., Giachetti D., Segura de Leon S. Global existence for parabolic problems involving the $p$-Laplacian and a critical gradient term // Indiana Univ. Math. J. 2009, V. 58, N 1. P. 1–48.
     
  19. Dall’Aglio A., De Cicco V., Giachetti D., Puel J.-P. Existence of bounded solutions for nonlinear elliptic equations in unbounded domains // NoDEA Nonlinear Differ. Equ. Appl. 2004. V. 11, N 4. P. 431–450.
     
  20. Nakao M., Chen C. Global existence and gradient estimates for the quasilinear parabolic equations of $m$-Laplacian type with a nonlinear convection term // J. Differ. Equ. 2000. V. 162, N 1. P. 224–250.
     
  21. Figueiredo D. G., Sanchez J., Ubilla P. Quasilinear equations with dependence on the gradient // Nonlinear Anal. Theory Meth. Appl. 2009. V. 71, N 10. P. 4862–4868.
     
  22. Iturriaga L., Lorca S., Sanchez J. Existence and multiplicity results for the $p$-Laplacian with a $p$-gradient term // NoDEA Nonlinear Differ. Equ. Appl. 2008. V. 15, N 6. P. 729–743.
     
  23. Li J., Yin J, Ke Y. Existence of positive solutions for the $p$-Laplacian with $p$-gradient term // J. Math. Anal. Appl. 2011. V. 383, N 1. P. 147–158.
     
  24. Ruiz D. A priori estimates and existence of positive solutions for strongly nonlinear problems // J. Differ. Equ. 2004. V. 199, N 1. P. 96–114.
     
  25. Zou H. H. A priori estimates and existence for quasi-linear elliptic equations // Calc. Var. Partial Differ. Equ. 2008. V. 33, N 4. P. 417–437.
     
  26. Dwivedi G., Gupta S. An existence result for $p$-Laplace equation with gradient nonlinearity in $\mathbb{R}^N$ // Commun. Math. 2022. V. 30, N 1. P. 149–159.
     
  27. Leonori T., Porretta A., Riey G. Comparison principles for $p$-Laplace equations with lower order terms // Ann. di Mat. Pura ed Appl. V. 196, N 3. P. 877–903.
     
  28. Bendahmane M., Karlsen K. H. Nonlinear anisotropic elliptic and parabolic equations in $\mathbb{R}^N$ with advection and lower order terms and locally integrable data // Potential Anal. 2005. V. 22, N 3. P. 207– 227.
     
  29. Fu Y., Pan N. Existence of solutions for nonlinear parabolic problem with $p(x)$-growth // J. Math. Anal. Appl. 2010. V. 362, N 2. P. 313–326.
     
  30. Zhan H. On anisotropic parabolic equations with a nonlinear convection term depending on the spatial variable // Adv. Differ. Equ. 2019. V. 2019, N 27. P. 1–26.
     
  31. Zhao J. Existence and nonexistence of solutions for $u_t = div(|\nabla u|^{p−2} \nabla u) + f(\nabla u, u, x, t)$ // J. Math. Anal. Appl. 1993. V. 172, N 1. P. 130–146.
     
  32. Tersenov Al. S., Tersenov Ar. S. Existence results for anisotropic quasilinear parabolic equations with time-dependent exponents and gradient term // J. Math. Anal. Appl. 2019. V. 480, N 1. P. 1–18.
     
  33. Терсенов Ар. С. Разрешимость задачи Дирихле для анизотропных параболических уравнений в невыпуклых областях // Сиб. журн. индустр. матем. 2022. Т. 25, № 1. С. 131–146.
     
  34. Терсенов Ар. С. О существовании вязких решений анизотропных параболических уравнений с переменным показателем анизотропности // Сиб. журн. индустр. матем. 2022. Т. 25, № 4. С. 206– 220.
     
  35. Tersenov Al. S., Tersenov Ar. S. The problem of Dirichlet for evolution one-dimensional $p$-Laplacian with nonlinear source // J. Math. An. Appl. 2008. V. 340, N 2. P. 1109–1119.
     
  36. Tersenov Al. S. The one-dimensional parabolic $p(x)$-Laplace equation // Nonlinear Differ. Equ. Appl. 2016. V. 23, N 27. P. 1–11.
     
  37. Wang L. On the regularity theory of fully nonlinear parabolic equation I // Comm. Pure. Appl. Math. 1992. V. 45. P. 27–76.
     
  38. Crandall M., Ishii H., Lions P.-L. User’s guide to viscosity solutions of second order partial differential equations // Bull. Amer. Math. Soc. 1992. V. 27, N 1. P. 1–67.
     
  39. Ладыженская О. А., Солонников В. А., Уральцева Н. Н. Линейные и квазилинейные уравнения параболического типа. М.: Наука, 1967.
     
  40. Gilding B. H. Hølder continuity of solutions of parabolic equations // J. London Math. Soc. 1976. V. 13, N 1. P. 103–106.
     
  41. Kruzhkov S. N. Quasilinear parabolic equations and systems with two independent variables // Trudy Sem. Petrovsk. 1979. V. 5. P. 217–272.

Работа выполнена в рамках государственного задания Института математики им. С. Л. Соболева СО РАН (проект FWNF-2022-0008). Других источников финансирования проведения или руководства данным конкретным исследованием не было.

А. С. Терсенов
  1. Институт математики им. С. Л. Соболева СО РАН, 
    просп. Акад. Коптюга, 4, г. Новосибирск 630090, Россия

E-mail: aterseno@math.nsc.ru

Статья поступила 06.11.2023 г.
После доработки — 17.09.2024 г.
Принята к публикации 06.11.2024 г.

Abstract:

In this paper, we study the first boundary value problem for $p(x)$-Laplacian with one spatial variable in the presence of gradient terms that do not satisfy the Bernstein—Nagumo condition. A class of gradient nonlinearities is defined, for which the existence of a viscosity solution that is Lipschitz continuous in $x$ and Hölder continuous in $t$ is proven.

References:
  1. E. Acerbi and G. Mingione, “Regularity results for stationary electro-rheological fluids,” Arch. Ration. Mech. Anal. 164, 213–259 (2002). 
     
  2. S. N. Antontsev and J. F. Rodrigues, “On stationary thermo-rheological viscous flows,” Ann. Univ. Ferrara, Sez. VII Sci. Mat. 52 (1), 19–36 (2006). 
     
  3. K. Rajagopal and M. Ru$\check{z}$i$\check{c}$ka, “Mathematical modelling of electro-rheological fluids,” Contin. Mech. Thermodyn. 13, 59–78 (2001). 
     
  4. M. Ru$\check{z}$i$\check{c}$ka, Electrorheological Fluids: Modeling and Mathematical Theory (Springer, Berlin, 2000). 
     
  5. R. Aboulaicha, D. Meskinea, and A. Souissia, “Regularity results for stationary electro-rheological fluids,” Arch. Ration. Mech. Anal. 164, 213–259 (2002). 
     
  6. Y. Chen, S. Levine, and M. Rao, “Variable exponent, linear growth functionals in image restoration,” SIAM J. Appl. Math. 66 (4), 1383–1406 (2006). 
     
  7. S. Antontsev and S. Shmarev, Evolution PDEs with Nonstandard Growth Conditions: Existence, Uniqueness, Localization, Blow-up (Atlantis Press, Paris, 2015). 
     
  8. R. Arora and S. Shmarev, “Strong solutions of evolution equations with $p(x, t)$-Laplacian: Existence, global higher integrability of the gradients and second-order regularity,” J. Math. Anal. Appl. 493 (1), 1–31 (2020). 
     
  9. R. Arora and S. Shmarev, “Existence and regularity results for a class of parabolic problems with double phase flux of variable growth,” Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. AMat. 117 (34), 1–48 (2023). 
     
  10. M. Belloni and B. Kawohl, “The pseudo-$p$-Laplace eigenvalue problem and viscosity solutions as $p \to \infty$,” ESAIM: Control Optim. Calc. Var. 10 (1), 28–52 (2004). 
     
  11. I. Birindelli and F. Demengel, “Existence and regularity results for fully nonlinear operators on the model of the pseudo Pucci’s operators,” J. Elliptic Parabol. Equat. 2, 171–187 (2016). 
     
  12. F. Demengel, “Lipschitz interior regularity for the viscosity and weak solutions of the pseudo $p$-Laplacian equation,” Adv. Differ. Equat. 21 (3), 373–400 (2016). 
     
  13. P. Juutinen, “On the definition of viscosity solutions for parabolic equations,” Proc. Am. Math. Soc. 129 (10), 2907–2911 (2001). 
     
  14. Ar. S. Tersenov, “Viscosity subsolutions and supersolutions for non-uniformly and degenerate elliptic equations,” Arch. Math. 45 (1), 19–35 (2009). 
     
  15. P. Juutinen, P. Lindqvist, and J. J. Manfredi, “On the equivalence of the viscosity solutions and weak solutions for a quasilinear equation,” SIAM J. Math. Anal. 33 (3), 699–717 (2001). 
     
  16. M. Medina and P. Ochoa, “On viscosity and weak solutions for non-homogeneous $p$-Laplace equations,” Adv. Nonlinear Anal. 8 (1), 468–481 (2019). 
     
  17. J. Siltakoski, “Equivalence of viscosity and weak solutions for a $p$-parabolic equation,” J. Evol. Equat. 21 (4), 2047–2080 (2021). 
     
  18. A. Dall’Aglio, D. Giachetti, and S. Segura de Leon, “Global existence for parabolic problems involving the $p$-Laplacian and a critical gradient term,” Indiana Univ. Math. J. 58 (1), 1–48 (2009). 
     
  19. A. Dall’Aglio, V. De Cicco, D. Giachetti, and J.-P. Puel, “Existence of bounded solutions for nonlinear elliptic equations in unbounded domains,” NoDEA Nonlinear Differ. Equat. Appl. 11 (4), 431–450 (2004). 
     
  20. M. Nakao and C. Chen, “Global existence and gradient estimates for the quasilinear parabolic equations of $m$-Laplacian type with a nonlinear convection term,” J. Differ. Equat. 162 (1), 224–250 (2000). 
     
  21. D. G. Figueiredo, J. Sanchez, and P. Ubilla, “Quasilinear equations with dependence on the gradient,” Nonlinear Anal. Theory Meth. Appl. 71 (10), 4862–4868 (2009). 
     
  22. L. Iturriaga, S. Lorca, and J. Sanchez, “Existence and multiplicity results for the $p$-Laplacian with a pgradient term,” NoDEA Nonlinear Differ. Equat. Appl. 15 (6), 729–743 (2008). 
     
  23. J. Li, J. Yin, and Y. Ke, “Existence of positive solutions for the $p$-Laplacian with $p$-gradient term,” J. Math. Anal. Appl. 383 (1), 147–158 (2011). 
     
  24. D. Ruiz, “A priori estimates and existence of positive solutions for strongly nonlinear problems,” J. Differ. Equat. 199 (1), 96–114 (2004). 
     
  25. H. H. Zou, “A priori estimates and existence for quasi-linear elliptic equations,” Calc. Var. Partial Differ. Equat. 33 (4), 417–437 (2008). 
     
  26. G. Dwivedi and S. Gupta, “An existence result for $p$-Laplace equation with gradient nonlinearity in $\mathbb {R}^N$ ,” Commun. Math. 30 (1), 149–159 (2022). 
     
  27. T. Leonori, A. Porretta, and G. Riey, “Comparison principles for $p$-Laplace equations with lower order terms,” Ann. Mat. Pura Appl. 196 (3), 877–903 (2017). 
     
  28. M. Bendahmane and K. H. Karlsen, “Nonlinear anisotropic elliptic and parabolic equations in $\mathbb {R}^N$ with advection and lower order terms and locally integrable data,” Potential Anal. 22 (3), 207–227 (2005). 
     
  29. Y. Fu and N. Pan, “Existence of solutions for nonlinear parabolic problem with $p(x)$-growth,” J. Math. Anal. Appl. 362 (2), 313–326 (2010). 
     
  30. H. Zhan, “On anisotropic parabolic equations with a nonlinear convection term depending on the spatial variable,” Adv. Differ. Equat. 2019 (27), 1–26 (2019). 
     
  31. J. Zhao, “Existence and nonexistence of solutions for $u_t = div(|\nabla u|^{p−2} \nabla u) + f(\nabla u, u, x, t)$,” J. Math. Anal. Appl. 172 (1), 130–146 (1993). 
     
  32. Al. S. Tersenov and Ar. S. Tersenov, “Existence results for anisotropic quasilinear parabolic equations with time-dependent exponents and gradient term,” J. Math. Anal. Appl. 480 (1), 1–18 (2019). 
     
  33. Ar. S. Tersenov, “Solvability of the Dirichlet problem for anisotropic parabolic equations in non-convex domains,” Sib. Zh. Ind. Mat. 25 (1), 131–146 (2022) [in Russian]. 
     
  34. Ar. S. Tersenov, “On the existence of viscous solutions of anisotropic parabolic equations with a variable anisotropy exponent,” Sib. Zh. Ind. Mat. 25 (4), 206–220 (2022) [in Russian]. 
     
  35. Al. S. Tersenov and Ar. S. Tersenov, “The problem of Dirichlet for evolution one-dimensional $p$-Laplacian with nonlinear source,” J. Math. An. Appl. 340 (2), 1109–1119 (2008). 
     
  36. Al. S. Tersenov, “The one-dimensional parabolic $p(x)$-Laplace equation,” Nonlinear Differ. Equat. Appl. 23 (27), 1–11 (2016). 
     
  37. L. Wang, “On the regularity theory of fully nonlinear parabolic equation I,” Commun. Pure. Appl. Math. 45, 27–76 (1992). 
     
  38. M. Crandall, H. Ishii, and P.-L. Lions, “User’s guide to viscosity solutions of second order partial differential equations,” Bull. Am. Math. Soc. 27 (1), 1–67 (1992). 
     
  39. O. A. Ladyzhenskaya, V. A. Solonnikov, and N. N. Ural’tseva, Linear and Quasilinear Equations of Parabolic Type (Nauka, Moscow, 1967) [in Russian]. 
     
  40. B. H. Gilding, “Hølder continuity of solutions of parabolic equations,” J. London Math. Soc. 13 (1), 103–106 (1976). 
     
  41. S. N. Kruzhkov, “Quasilinear parabolic equations and systems with two independent variables,” Tr. Semin. Petrovskogo 5, 217–272 (1979) [in Russian].