On topological instabilities arising in families of semilinear parabolic problems

It is known that for a broad class of one-parameter families of semilinear elliptic problems, a discontinuity in the branch of minimal solutions can be induced by arbitrarily small perturbations of the nonlinearity, when the spatial dimension is equal to one or two. In this article, we communicate, from a topological point of view, on the dynamical implications of such a sensitivity of the minimal branch, for the corresponding one-parameter family of semilinear parabolic problems. The key ingredients to do so, rely on a combination of a general continuation result from the Leray-Schauder degree theory regarding the existence of an unbounded continuum of solutions to one-parameter families of elliptic problems, and a growth property of the branch of minimal solutions to such problems. In particular, it is shown that the phase portrait of the semigroup associated with $\partial_t u -\Delta u=\lambda g(u)$, can experience a topological instability, when the function $g$ is locally perturbed. More precisely, it is shown that for all $\epsilon >0$, there exists $\widehat{g}$ such that $\|g-\widehat{g}\|_{\infty} \leq \epsilon$ ($g-\widehat{g}$ being with compact support) for which the semigroup associated with $\partial_t u -\Delta u=\lambda \widehat{g}(u)$ possesses multiple equilibria, for certain $\lambda$-values at which the semigroup associated with $\partial_t u -\Delta u=\lambda g(u)$ possesses only one equilibrium. The mechanism at the origin of such an instability is also clarified. The latter results from a local deformation of the $\lambda$-bifurcation diagram (associated with $-\Delta u=\lambda g(u)$, $u\vert_{\partial \Omega}=0$) by the creation of a multiple-point or a new fold-point on it when a small perturbation is applied. This is proved under assumptions on $g$ that prevents the use of linearization techniques.