Nonlinear parabolic equations with blowing-up coefficients with respect to the unknown and with soft measure data

Abstract

We establish the existence of solutions for the nonlinear parabolic problem with Dirichlet homogeneous boundary conditions, $$ \frac{\partial u}{\partial t} - \sum_{i=1}^N\frac{\partial}{\partial x_i} \Big( d_i(u)\frac{\partial u}{\partial x_i} \Big) =\mu,\quad u(t=0)=u_0, $$ in a bounded domain. The coefficients $d_i(s)$ are continuous on an interval $]-\infty,m[$, there exists an index p such that $d_p(u)$ blows up at a finite value m of the unknown u, and $\mu$ is a diffuse measure.

Keywords

• Nonlinear parabolic equations
• blowing-up coefficients
• renormalized solutions
• soft measure