On the stability of solutions of semilinear elliptic equations with Robin boundary conditions on Riemannian manifolds

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

Reaction-diffusion problems on time-dependent Riemannian manifolds: stability of periodic solutions

We investigate the stability of time-periodic solutions of semilinear parabolic problems with Neumann boundary conditions. Such problems are posed on compact submanifolds evolving periodically in time. The discussion is based on the principal eigenvalue of periodic parabolic operators. The study is motivated by biological models on the effect of growth and curvature on patterns formation. The Ricci curvature plays an important role

A note on existence of patterns on surfaces of revolution with nonlinear flux on the boundary

In this note we address the question of existence of non-constant stable stationary solution to the heat equation on surfaces of revolution subject to nonlinear boundary flux involving a positive parameter. Our result is independent of the surface geometry and, in addition, we provide the asymptotic profile of the solutions and some examples where the result applies

Stability and instability of solutions of semilinear problems with Dirichlet boundary condition on surfaces of revolution

We consider the equation $\Delta u+f(u)=0$ on a surface of revolution with Dirichlet boundary conditions. We obtain conditions on $f$, the geometry of the surface and the maximum value of a positive solution in order to ensure its stability or instability. Applications are given for our main results

Fast diffusion on noncompact manifolds: well-posedness theory and connections with semilinear elliptic equations

We investigate the well-posedness of the fast diffusion equation (FDE) in a wide class of noncompact Riemannian manifolds. Existence and uniqueness of solutions for globally integrable initial data was established in [5]. However, in the Euclidean space, it is known from Herrero and Pierre [20] that the Cauchy problem associated with the FDE is well posed for initial data that are merely in $L^1_{\mathrm{loc}}$. We establish here that such data still give rise to global solutions on general Riemannian manifolds. If, in addition, the radial Ricci curvature satisfies a suitable pointwise bound from below (possibly diverging to $-\infty$ at spatial infinity), we prove that also uniqueness holds, for the same type of data, in the class of strong solutions. Besides, under the further assumption that the initial datum is in $L^2_{\mathrm{loc}}$ and nonnegative, a minimal solution is shown to exist, and we are able to establish uniqueness of purely (nonnegative) distributional solutions, which to our knowledge was not known before even in the Euclidean space. The required curvature bound is in fact sharp, since on model manifolds it turns out to be equivalent to stochastic completeness, and it was shown in [13] that uniqueness for the FDE fails even in the class of bounded solutions on manifolds that are not stochastically complete. Qualitatively this amounts to asking that the curvature diverges at most quadratically at infinity. A crucial ingredient of the uniqueness result is the proof of nonexistence of distributional subsolutions to certain semilinear elliptic equations with power nonlinearities, of independent interest

New eigenvalue estimates involving Bessel functions

Given a compact Riemannian manifold (Mn, g) with boundary ∂M, we give an estimate for the quotient R ∂M f dµg R M f dµg , where f is a smooth positive function defined on M that satisfies some inequality involving the scalar Laplacian. By the mean value lemma established in [39], we provide a differential inequality for f which, under some curvature assumptions, can be interpreted in terms of Bessel functions. As an application of our main result, a new inequality is given for Dirichlet and Robin Laplacian. Also, a new estimate is established for the eigenvalues of the Dirac operator that involves a positive root of Bessel function besides the scalar curvature. Indepen[1]dently, we extend the Robin Laplacian on functions to differential forms. We prove that this natural extension defines a self-adjoint and elliptic operator whose spectrum is discrete and consists of positive real eigenvalues. In particular, we characterize its first eigenvalue and provide a lower bound of it in terms of Bessel functions