Propagation and extinction of fronts for semilinear parabolic equations on Riemannian manifolds
In this article, the author considers the initial value problem for a semilinear parabolic equation of the form \(\frac{\partial u}{\partial t}= \Delta u + f(u)\) in \(M\times (0,+\infty), u = u_0 \) in \(M\times \{0\} \) in the following framework: \(M\) is an \(n\)-dimensional complete, non compact Riemanian manifold, \(f\in C^1([0,1])\), \(f(0) = f(1) =0,\) \(u_0\) is continuous on \(M\) and satifies \(0 \leq u_0(x) \leq 1.\) In this context, a unique global in time solution exists. The question discussed in this paper is then to give conditions on \(M,f\) in order that the solution satisfies either \(\lim_{t\to + \infty} u(t,x) = 0\) (extinction) or \(\lim_{t\to + \infty} u(t,x) = 1\) (propagation). The main results of the paper give several very technical sufficient conditions on the manifold (through the sectional curvature, the Ricci curvature\dots) for the above properties to hold.
scientific article; zbMATH DE number 6521863
scientific article