Young measure solutions of a class of forward--backward diffusion equations
In this paper, the authors consider the problem \[ u_t= \nabla\cdot q(\nabla u),\quad (x,t)\in \Omega\times (0,T),\tag{1} \] where \(\Omega \subset {\mathbb R}^N\), is a bounded domain with smooth boundary, with Dirichlet boundary conditions on \(\partial \Omega\) and appropriate initial conditions. The assumptions on \(q\) are that \(q = \nabla \varphi\), where \(\varphi \in C^1(\mathbb R^N)\), \[ (\lambda |A|^{1+\delta} -1)_+ \leq \varphi(A) \leq \Lambda |A|^{1-\delta} + 1 \] and \(|q(A)|\leq \Lambda |A|^\delta\) for all \(A \in {\mathbb R}^N\), \(\delta \in [0,1]\). These assumptions cover a multitude of important cases and include backward-forward diffusion and equations of interest in the theory of phase transitions. The main results of the paper are contained in Theorems 3.1 and 4.1 and complement the results of \textit{S. Demoulini} [SIAM J. Math. Anal. 27, 376-403 (1996; Zbl 0851.35066)], which are only valid in the Hilbert space case \(\delta=1\). Theorem 3.1 deals with the regular case \(\delta \in (0,1)\) and the result is that if the initial data \(u_0 \in W_0^{1,1+\delta}(\Omega) \cap L^\infty (\Omega)\), equation (1) with Dirichlet boundary conditions admits at least one Young measure solution. Similarly, in the singular case \(\delta=0\), we have that if \(u_0\in BV(\Omega) \cap L^\infty (\Omega)\) and satisfies the boundary condition in the sense of trace, (1) admits at least one Young measure solution. There are many components in the proofs, but the main idea in Theorem 3.1 is [following \textit{D. Kinderlehrer} and \textit{P. Pedregal}, SIAM J. Math. Anal. 23, 1-19 (1992; Zbl 0757.49014)] to prove existence of Young measure solutions to time difference equations and then pass to the limit, while in Theorem 4.1 one takes a limit of solutions in the regular case as \(\delta\) goes to zero.
scientific article; zbMATH DE number 1915778
scientific article