# Well-posedness results for triply nonlinear degenerate parabolic equations - Mathematics > Analysis of PDEs

Well-posedness results for triply nonlinear degenerate parabolic equations - Mathematics > Analysis of PDEs - Download this document for free, or read online. Document in PDF available to download.

Abstract: We study the well-posedness of triply nonlinear degenerateelliptic-parabolic-hyperbolic problem $$ bu t - { m div} \tilde{\mathfraka}u, abla\phiu+\psiu=f, \quad u| {t=0}=u 0 $$ in a bounded domain withhomogeneous Dirichlet boundary conditions. The nonlinearities $b,\phi$ and$\psi$ are supposed to be continuous non-decreasing, and the nonlinearity$\tilde{\mathfrak a}$ falls within the Leray-Lions framework. Some restrictionsare imposed on the dependence of $\tilde{\mathfrak a}u, abla\phiu$ on $u$and also on the set where $\phi$ degenerates. A model case is $\tilde{\mathfraka}u, abla\phiu=\tilde{\mathfrak{f}}bu,\psiu,\phiu+ku\mathfrak{a} 0 abla\phiu,$with $\phi$ which is strictly increasing except on a locally finite number ofsegments, and $\mathfrak{a} 0$ which is of the Leray-Lions kind. We areinterested in existence, uniqueness and stability of entropy solutions. If$b=\mathrm{Id}$, we obtain a general continuous dependence result on data$u 0,f$ and nonlinearities $b,\psi,\phi,\tilde{\mathfrak{a}}$. Similar resultis shown for the degenerate elliptic problem which corresponds to the case of$b\equiv 0$ and general non-decreasing surjective $\psi$. Existence, uniquenessand continuous dependence on data $u 0,f$ are shown when $b+\psi\R=\R$ and$\phi\circ b+\psi^{-1}$ is continuous.

Author: ** Boris Andreianov, Mostafa Bendahmane, Kenneth K. Karlsen, Stanislas Ouaro**

Source: https://arxiv.org/