On a Class of Forward-Backward Parabolic Equations: Properties of Solutions

Source abstract

We study the equation $u_t=[\varphi(u)]_{xx}+\varepsilon [\psi(u)]_{txx}$ with suitable boundary conditions and a nonnegative Radon measure as initial datum. Here $\varphi(0)=\varphi(\infty)=0$, $\varphi$ is increasing in $(0,\alpha)$ and decreasing in $(\alpha,\infty)$, and the regularizing term $\psi$ is increasing but bounded. It is natural to study measure-valued solutions since singularities may appear spontaneously in finite time. Positive measure-valued solutions are known to exist and to not be unique. In this paper we study qualitative properties shared by all solutions of the problem. We prove, among other things, that the singular part of a solution is nondecreasing with respect to time, so its support is nonshrinking, and, due to the possible appearance of singularities, may even expand. This phenomenon sharply distinguishes the case of bounded $\psi$ from those of power-type $\psi$, where the singular part remains constant in time, and logarithmic $\psi$, where the singular part may grow but its support does not expand. It also distinguishes the present case from the case of $\varphi$ increasing in $(0,\alpha)$, decreasing in $(\alpha,\beta)$, increasing in $(\beta,\infty)$ for some $0<\alpha<\beta<\infty$, and bounded (with $\psi$ as in this paper), where the singular part of a solution is nonincreasing in time and singularities may disappear.