Research article

Stability properties of Radon measure-valued solutions for a class of nonlinear parabolic equations under Neumann boundary conditions

  • Received: 13 June 2021 Accepted: 19 August 2021 Published: 24 August 2021
  • MSC : 35K65, 35K61, 35B40, 28A33, 35R06, 28A50

  • 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

    Related Papers:

  • 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.



    加载中


    [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
  • Reader Comments
  • © 2021 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(2242) PDF downloads(101) Cited by(1)

Article outline

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog