Abstract
We consider the porous medium equation with a power-like reaction term, posed on Riemannian manifolds. Under certain assumptions on p and m in (1.1), and for small enough nonnegative initial data, we prove existence of global in time solutions, provided that the Sobolev inequality holds on the manifold. Furthermore, when both the Sobolev and the Poincaré inequalities hold, similar results hold under weaker assumptions on the forcing term. By the same functional analytic methods, we investigate global existence for solutions to the porous medium equation with source term and variable density in {{mathbb {R}}}^n.
Highlights
We investigate existence of global in time solutions to nonlinear reaction–diffusion problems of the following type: ut = um + u p in M × (0, T )
(See Theorem 2.2) We prove global existence of solutions to (1.1), assuming that the initial datum is sufficiently small, that p>m+ 2, N
(See Theorem 2.5) We show that, if both the Sobolev and the Poincaré inequalities (i.e. (1.2), (1.3)) hold, for any p > m, for any sufficiently small initial datum u0, belonging to suitable Lebesgue spaces, there exists a global solution u(t) such that u(t) ∈ L∞(M)
Summary
We investigate existence of global in time solutions to nonlinear reaction–diffusion problems of the following type: ut = um + u p in M × (0, T ). We shall assume throughout this paper that. Let Lq (M) be the space of those measurable functions f such that | f |q is integrable w.r.t. the Riemannian measure μ. We shall always assume that M supports the Sobolev inequality, namely that:. Mathematics Subject Classification: Primary: 35K57, Secondary: 35B44, 58J35, 35K65, 35R01 Keywords: Reaction–diffusion equations, Riemannian manifolds, Blow-up, Global existence, Diffusions with weights. In one of our main results, we shall suppose that M supports the Poincaré inequality, namely that:. As is well known, in RN (1.2) holds, but (1.3) fails, whereas on the hyperbolic space both (1.2) and (1.3) are fulfilled
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