Existence and asymptotic behavior of solutions to a nonlinear parabolic equation of fourth order
In this work the existence of weak solutions of the parabolic partial differential equation \(u_t+ \nabla\cdot (|\nabla \Delta u|^{p-2}\nabla\Delta u)= f(u)\) with boundary conditions \(u=\Delta u=0\) and initial condition \(u_0\) is analyzed. For the stationary solutions, a functional minimization and Leray-Schauder's fixed point theorem lead to the existence statement. For the developing case semi-discrete problems are solved and the solutions are used to define a second approximate solution. Both tend in the limit to the sought unique weak solution. This is shown by determination suitable bounds on the derivatives in time and space, and using a compactness argument. In the end of this work it is shown that in the \(p\to \infty\) limit the solutions converge to the initial data.
scientific article; zbMATH DE number 5349083
scientific article