We study an evolution problem corresponding to the nonlinear diffusion equation $u_t = \Delta \varphi (u) + {\operatorname{div}}(u{\operatorname {grad}}v)$ with no flux boundary conditions. This problem has a continuum of stationary solutions. We prove the existence and uniqueness of the solution of the evolution problem and construct a Lyapunov functional in order to show that the solution stabilizes as $t \to \infty $.
Bertsch, M., Hilhorst, D. (1986). A density dependent diffusion equation in population dynamics: stabilization to equilibrium. SIAM JOURNAL ON MATHEMATICAL ANALYSIS, 17(4), 863-883 [10.1137/0517062].
A density dependent diffusion equation in population dynamics: stabilization to equilibrium
BERTSCH, MICHIEL;
1986-01-01
Abstract
We study an evolution problem corresponding to the nonlinear diffusion equation $u_t = \Delta \varphi (u) + {\operatorname{div}}(u{\operatorname {grad}}v)$ with no flux boundary conditions. This problem has a continuum of stationary solutions. We prove the existence and uniqueness of the solution of the evolution problem and construct a Lyapunov functional in order to show that the solution stabilizes as $t \to \infty $.File in questo prodotto:
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