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Register a new userOn the stability of solutions of semilinear elliptic equations with Robin boundary conditions on Riemannian manifolds
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We investigate existence and nonexistence of stationary stable nonconstant solutions, i.e. patterns, of semilinear parabolic problems in bounded domains of Riemannian manifolds satisfying Robin boundary conditions. These problems arise in several models in applications, in particular in Mathematical Biology. We point out the role both of the nonlinearity and of geometric objects such as the Ricci curvature of the manifold, the second fundamental form of the boundary of the domain and its mean curvature. Special attention is devoted to surfaces of revolution and to spherically symmetric manifolds, where we prove refined results.
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We study existence and non-existence of global solutions to the semilinear heat equation with a drift term and a power-like source term, on Cartan-Hadamard manifolds. Under suitable assumptions on Ricci and sectional curvatures, we show that global solutions cannot exists if the initial datum is large enough. Furthermore, under appropriate conditions on the drift term, global existence is obtained, if the initial datum is sufficiently small. We also deal with Riemannian manifolds whose Ricci curvature tends to zero at infinity sufficiently fast.
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