JournalsaihpcVol. 36, No. 5pp. 1431–1465

Parabolic equation in time and space dependent anisotropic Musielak–Orlicz spaces in absence of Lavrentiev's phenomenon

  • Iwona Chlebicka

    Institute of Mathematics, Polish Academy of Sciences, ul. Śniadeckich 8, 00-656 Warsaw, Poland, Institute of Applied Mathematics and Mechanics, University of Warsaw, ul. Banacha 2, 02-097 Warsaw, Poland
  • Piotr Gwiazda

    Institute of Mathematics, Polish Academy of Sciences, ul. Śniadeckich 8, 00-656 Warsaw, Poland
  • Anna Zatorska–Goldstein

    Institute of Applied Mathematics and Mechanics, University of Warsaw, ul. Banacha 2, 02-097 Warsaw, Poland
Parabolic equation in time and space dependent anisotropic Musielak–Orlicz spaces in absence of Lavrentiev's phenomenon cover
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Abstract

We study a general nonlinear parabolic equation on a Lipschitz bounded domain in RN\mathbb{R}^{N},

{tudivA(t,x,u)=f(t,x)inΩT,u(t,x)=0on(0,T)×Ω,u(0,x)=u0(x)inΩ,\left\{\begin{matrix} \partial _{t}u−\mathrm{div}A(t,x,\mathrm{∇}u) = f(t,x)\: & \text{in}\:\:\mathrm{\Omega }_{T}, \\ u(t,x) = 0\: & \text{on}\:(0,T) \times \partial \mathrm{\Omega }, \\ u(0,x) = u_{0}(x)\: & \text{in}\:\mathrm{\Omega }, \\ \end{matrix}\right.

with fL(ΩT)f \in L^{\infty }(\mathrm{\Omega }_{T}) and u0L(Ω)u_{0} \in L^{\infty }(\mathrm{\Omega }). The growth of the monotone vector field A is controlled by a generalized fully anisotropic N-function M:[0,T \times \mathrm{\Omega } \times \mathbb{R}^{N}\rightarrow [0,\infty \right.\right. inhomogeneous in time and space, and under no growth restrictions on the last variable. It results in the need of the integration by parts formula which has to be formulated in an advanced way. Existence and uniqueness of solutions are proven when the Musielak–Orlicz space is reflexive OR in absence of Lavrentiev's phenomenon. To ensure approximation properties of the space we impose natural assumption that the asymptotic behaviour of the modular function is sufficiently balanced. Its instances are log-Hölder continuity of variable exponent or optimal closeness condition for powers in double phase spaces.

The noticeable challenge of this paper is considering the problem in non-reflexive and inhomogeneous fully anisotropic space that changes along time.

Cite this article

Iwona Chlebicka, Piotr Gwiazda, Anna Zatorska–Goldstein, Parabolic equation in time and space dependent anisotropic Musielak–Orlicz spaces in absence of Lavrentiev's phenomenon. Ann. Inst. H. Poincaré Anal. Non Linéaire 36 (2019), no. 5, pp. 1431–1465

DOI 10.1016/J.ANIHPC.2019.01.003