In this paper, we address the existence, uniqueness, decay estimates, and the large-time behavior of the Radon measure-valued solutions for a class of nonlinear strongly degenerate parabolic equations involving a source term under Neumann boundary conditions with bounded Radon measure as initial data.
$ \begin{equation*} \begin{cases} u_{t} = \Delta\psi(u)+h(t)f(x, t) \ \ &\text{in} \ \ \Omega\times(0, T), \\ \frac{\partial\psi(u)}{\partial\eta} = g(u) \ \ &\text{on} \ \ \partial\Omega\times(0, T), \\ u(x, 0) = u_{0}(x) \ \ &\text{in} \ \ \Omega, \end{cases} \end{equation*} $
where $ T > 0 $, $ \Omega\subset \mathbb{R}^{N}(N\geq2) $ is an open bounded domain with smooth boundary $ \partial\Omega $, $ \eta $ is an outward normal vector on $ \partial\Omega $. The initial value data $ u_{0} $ is a nonnegative bounded Radon measure on $ \Omega $, the function $ f $ is a solution of the linear inhomogeneous heat equation under Neumann boundary conditions with measure data, and the functions $ \psi $, $ g $ and $ h $ satisfy the suitable assumptions.
Citation: Quincy Stévène Nkombo, Fengquan Li, Christian Tathy. Stability properties of Radon measure-valued solutions for a class of nonlinear parabolic equations under Neumann boundary conditions[J]. AIMS Mathematics, 2021, 6(11): 12182-12224. doi: 10.3934/math.2021707
In this paper, we address the existence, uniqueness, decay estimates, and the large-time behavior of the Radon measure-valued solutions for a class of nonlinear strongly degenerate parabolic equations involving a source term under Neumann boundary conditions with bounded Radon measure as initial data.
$ \begin{equation*} \begin{cases} u_{t} = \Delta\psi(u)+h(t)f(x, t) \ \ &\text{in} \ \ \Omega\times(0, T), \\ \frac{\partial\psi(u)}{\partial\eta} = g(u) \ \ &\text{on} \ \ \partial\Omega\times(0, T), \\ u(x, 0) = u_{0}(x) \ \ &\text{in} \ \ \Omega, \end{cases} \end{equation*} $
where $ T > 0 $, $ \Omega\subset \mathbb{R}^{N}(N\geq2) $ is an open bounded domain with smooth boundary $ \partial\Omega $, $ \eta $ is an outward normal vector on $ \partial\Omega $. The initial value data $ u_{0} $ is a nonnegative bounded Radon measure on $ \Omega $, the function $ f $ is a solution of the linear inhomogeneous heat equation under Neumann boundary conditions with measure data, and the functions $ \psi $, $ g $ and $ h $ satisfy the suitable assumptions.
[1] | J. L. Vázquez, Measure-valued solutions and the phenomenon of blow-down in logarithmic diffusion, J. Math. Anal. Appl., 352 (2009), 515-547. |
[2] | S. Itǒ, Fundamental solutions of parabolic differential equations and boundary value problems, Jnp. J. Math., 27 (1957), 55-102. |
[3] | S. Itǒ, A boundary value problem of partial differential equations of parabolic type, Duke Math. J., 24 (1957), 299-312. |
[4] | C. S. Kahane, On the asymptotic behavior of solutions of parabolic equations, Czechoslovak Math. J., 33 (1983), 262-285. doi: 10.21136/CMJ.1983.101876 |
[5] | C. V. Pao, Nonlinear Parabolic and Elliptic Equations, New York: Plenum Press, 1992. |
[6] | M. Bertsch, F. Smarrazzo, A. Tesei, Pseudo-parabolic regularization of forward-backward parabolic equations: Power-type nonlinearities, J. Reine Angew. Math., 712 (2016), 51-80. |
[7] | M. Bertsch, F. Smarrazzo, A. Tesei, Pseudo-parabolic regularization of forward-backward parabolic equations: A logarithmic nonlinearity, Anal. PDE, 6 (2013), 1719-1754. doi: 10.2140/apde.2013.6.1719 |
[8] | L. C. Evance, Partial differential equations, Graduate Studies in Mathematics, American Mathematical Society, Providence, R.I., 2010. |
[9] | M. M. Porzio, F. Smarrazzo, A. Tesei, Radon measure-valued solutions for a class of quasilinear parabolic equations, Arch. Ration. Mech. Anal., 210 (2013), 713-772. doi: 10.1007/s00205-013-0666-0 |
[10] | M. M. Porzio, F. Smarrazzo, A. Tesei, Radon measure-valued solutions of nonlinear strongly degenerate parabolic equations, Calc. Var. Partial Dif., 51 (2014), 401-437. doi: 10.1007/s00526-013-0680-y |
[11] | F. Smarrazzo, A. Tesei, Degenerate regularization of forward-backward parabolic equations: The regularized problem, Arch. Ration. Mech. Anal., 204 (2012), 85-139. doi: 10.1007/s00205-011-0470-7 |
[12] | L. Orsina, M. M. Porzio, F. Smarrazzo, Measure-valued solutions of nonlinear parabolic equations with logarithmic diffusion, J. Evol. Equ., 15 (2015), 609-645. doi: 10.1007/s00028-015-0275-5 |
[13] | M. M. Porzio, F. Smarrazzo, Radon measure-valued solutions for some quasilinear degenerate elliptic equations, Ann. Mat. Pura Appl., 194 (2015), 495-532. doi: 10.1007/s10231-013-0386-y |
[14] | F. Petitta, A. C. Ponce, A. Porretta, Approximation of diffuse measures for parabolic capacities, C. R. Math. Acad. Sci. Paris, 346 (2008), 161-166. doi: 10.1016/j.crma.2007.12.002 |
[15] | J. Droniou, A. Porretta, A. Prignet, Parabolic capacity and soft measures for nonlinear equations, Potential Anal., 19 (2003), 99-161. doi: 10.1023/A:1023248531928 |
[16] | F. Petitta, A. C. Ponce, A. Porretta, Diffuse measures and nonlinear parabolic equations, J. Evol. Equ., 11 (2011), 861-905. doi: 10.1007/s00028-011-0115-1 |
[17] | J. L. Vázquez, The Porous Medium Equation: Mathematical Theory, Oxford: Oxford Mathematical Monographs, 2007. |
[18] | J. R. Anderson, Local existence and uniqueness of solutions of degenerate parabolic equations, Commun. Part. Diff. Eq., 16 (1991), 105-143. doi: 10.1080/03605309108820753 |
[19] | E. DiBenedetto, Continuity of weak solutions to a general porous medium equation, Indiana Univ. Math. J., 32 (1983), 83-118. doi: 10.1512/iumj.1983.32.32008 |
[20] | J. Ding, B. Z. Guo, Global existence and blow-up solutions for quasilinear reaction-diffusion equations with a gradient term, Appl. Math. Lett., 24 (2011), 936-942. doi: 10.1016/j.aml.2010.12.052 |
[21] | J. L. Lions, Quelques Méthodes de Résolutions des Problèmes aux Limites non Linéaires, Paris: Dunod., French, 1969. |
[22] | O. A. Ladyženskaja, V. A. Solonnikov, N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type, Translations of Mathematical Monographs, American Mathematical Society, Providence, R.I., 23 (1968). |
[23] | J. Simon, Compact sets in the space $L^{p}(0, T;B)$, Ann. Mat. Pura Appl., 146 (1987), 65-96. |
[24] | L. C. Evans, R. F. Gariepy, Measure Theory and Fine Properties of Functions, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1992. |
[25] | M. Papi, M. M. Porzio, F. Smarrazzo, Existence of solutions to a class of weakly coercive diffusion equations with singular initial data, Adv. Differential Eq., 22 (2017), 893-963. |
[26] | L. Boccardo, L. Moreno-Mérida, $W^{1, 1}(\Omega)$ Solutions of nonlinear problems with nonhomogeneous Neumann boundary conditions, Milan J. Math., 83 (2015), 279-293. doi: 10.1007/s00032-015-0235-0 |
[27] | E. DiBenedetto, Degenerate Parabolic Equations, Universitext, New York: Springer-Verlag, 1993. |
[28] | H. Amann, Quasilinear parabolic system under nonlinear boundary conditions, Arch. Rational Mech. Anal., 92 (1986), 53-192. |
[29] | H. Amann, Parabolic evolution equations and nonlinear boundary conditions, J. Differ. Equations, 73 (1988), 201-269. |
[30] | L. Dung, Remarks on Hölder continuity for parabolic equations and convergence to global attractors, Nonlinear Anal., 41 (2000), 921-941. doi: 10.1016/S0362-546X(98)00319-8 |
[31] | V. Bögelein, F. Duzaar, U. Gianazza, Porous medium type equations with measure data and potential estimates, SIAM J. Math. Anal., 45 (2013), 3283-3330. doi: 10.1137/130925323 |
[32] | V. Liskevich, I. I. Skrypnik, Pointwise estimates for solutions to the porous medium equation with measure as a forcing term, Israel J. Math., 194 (2013), 259-275. doi: 10.1007/s11856-012-0098-9 |
[33] | U. Gianazza, Degenerate and singular porous medium type equations with measure data, Elliptic and Parabolic Equations, 119 (2015), 139-158. |
[34] | H. Brézis, W. A. Strauss, Semi-linear second-order elliptic equations in $L^{1}$, J. Math. Soc. Japan, 25 (1973), 565-590. |
[35] | J. R. Cannon, The One-dimensional Heat Equations, Encyclopedia of Mathematics and its Applications, Addison-Wesley Publishing Company, 23 (1984). |
[36] | G. Q. Chen, H. Frid, Extended divergence measure fields and the Euler equations for gas dynamic, Comm. Math, Phys., 236 (2003), 251-280. doi: 10.1007/s00220-003-0823-7 |
[37] | B. P. Andreianov, N. Igbida, Uniqueness for inhomogeneous Dirichlet problem for elliptic-parabolic equations, Proc. Roy. Soc. Edinburgh Sect. A, 137 (2007), 1119-1133 doi: 10.1017/S0308210505001290 |
[38] | L. Boccardo, J. M. Mazón, Existence of solutions and regularizing effect for some elliptic nonlinear problems with nonhomogeneous Neumann boundary conditions, Rev. Mat. Complut., 28 (2015), 263-280. doi: 10.1007/s13163-014-0162-6 |
[39] | F. Audreu, N. Igbida, J. M. Mazón, J. Toledo, Degenerate elliptic equations with nonlinear boundary conditions and measure data, Ann. Sc. Norm. Super. Pisa. Cl. Sci., 8 (2009), 769-803. |
[40] | F. Audreu, J. M. Mazón, S. Segura De Léon, J. Toledo, Quasilinear elliptic and parabolic equations in $L^{1}$ with nonlinear boundary conditions, Adv. Math. Sci. Appl., 7 (1997), 183-213. |
[41] | M. M. Porzio, F. Smarrazzo, Radon measure-valued solutions for some quasilinear degenerate elliptic equations, Ann. Mat. Pura Appl., 194 (2015), 495-532. doi: 10.1007/s10231-013-0386-y |
[42] | S. Itô, The fundamental solution of the parabolic equation in a differentiable manifold. II, Osaka Math. J., 6 (1954), 167-185. |
[43] | A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall, Inc., Englewood Cliffs, N. J., 1964. |
[44] | M. Giaquinta, G. Modica, J. Souček, Cartesian Currents in the Calculus of Variations. I. Cartesian Currents, Berlin: Springer-Verlag, 1998. |
[45] | M. Bertsch, F. Smarrazzo, A. Terracina, A. Tesei, Radon measure-valued solutions of first order scalar conservation laws, Adv. Nonlinear Anal., 9 (2020), 65-107. |
[46] | Q. S. Nkombo, F. Li, Radon Measure-valued solutions for nonlinear strongly degenerate parabolic equations with measure data, Eur. J. Pure Appl. Math., 14 (2021), 204-233. doi: 10.29020/nybg.ejpam.v14i1.3877 |
[47] | U. S. Fjordholm, S. Mishra, E. Tadmor, On the computation of measure-valued solutions, Acta Numer., 25 (2016), 567-679. doi: 10.1017/S0962492916000088 |
[48] | S. Lanthaler, S. Mishra, Computation of measure-valued solutions for the incompressible Euler equations, Math. Models Methods Appl. Sci., 25 (2015), 2043-2088. doi: 10.1142/S0218202515500529 |
[49] | J. Droniou, T. Gallouët, R. Herbin, A finite volume scheme for a noncoercive elliptic equation with measure data, SIAM J. Numer. Anal., 41 (2003), 1997-2031. |
[50] | R. Eymard, T. Gallouët, R. Herbin, A. Michel, Convergence of a finite volume scheme for nonlinear degenerate parabolic equations, Numer. Math., 92 (2002), 41-82. doi: 10.1007/s002110100342 |