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Parabolic equation in time and space dependent anisotropic Musielak-Orlicz spaces in absence of Lavrentiev’s phenomenon.(English)Zbl 1419.35092

The authors study a general nonlinear parabolic equation on a Lipschitz bounded domain with bounced and measurable data:We study a general nonlinear parabolic equation on a Lipschitz bounded domain in \(\mathbb{R}^N\),\[\begin{cases} \partial_t u - \operatorname{div} A(t, x, \nabla u) = f(t, x) \quad & \text{in } \Omega_T, \\u(t, x) = 0 & \text{on }(0, T) \times \partial \Omega, \\ u(0, x) = u_0(x) & \text{in } \Omega, \end{cases}\]with \(f \in L^\infty(\Omega_T)\) and \(u_0 \in L^\infty(\Omega)\). The growth of the monotone vector field \(A(t,x,\nabla u) \) is controlled by a generalized fully anisotropic \(N\)-function \(M : [0, T) \times \Omega \times \mathbb{R}^N \rightarrow [0, \infty)\) inhomogeneous in time and space, and under no growth restrictions on the last variable. Existence and uniqueness of solutions are proved when the Musielak-Orlicz space is reflexive or in absence of Lavrentiev’s phenomenon. Notably challenging is the problem in non-reflexive and inhomogeneous fully anisotropic space that changes along time. To get this so interesting results, the authors impose natural assumption on the asymptotic behaviour of the modular function.

MSC:

35K55 Nonlinear parabolic equations
35A01 Existence problems for PDEs: global existence, local existence, non-existence
35B65 Smoothness and regularity of solutions to PDEs

Cite

References:

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This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
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