Stabilizing Effect of Diffusion and Dirichlet Boundary Conditions
Vol. 17 (2010), No. 3, Page 303--312.
Bud\'a\v c, Ondrej ;Fila, Marek
Stabilizing Effect of Diffusion and Dirichlet Boundary Conditions
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Abstract:
It is known that diffusion together with Dirichlet boundary conditions can inhibit the occurrence of blow-up. We examine the question how strong is this stabilizing effect for reaction-diffusion equations in one space-dimension. We show that if all positive solutions of an ODE blow up in finite time then for the corresponding parabolic PDE (obtained by adding diffusion and the Dirichlet boundary condition) there is either an unbounded sequence of stationary solutions or an unbounded time-dependent solution.
Keywords: Blow-up, reaction-diffusion, stabilization of solutions.
Mathematics Subject Classification (2010): Primary 35K57, Secondary 35B35.
Received: 2010-09-17
