Well-posedness results for triply nonlinear degenerate parabolic equations

Andreianov, Boris; Bendahmane, Mostafa; Karlsen, Kenneth K.; Ouaro, Stanislas

Well-posedness results for triply nonlinear degenerate parabolic equations

Authors

Description

We study the well-posedness of triply nonlinear degenerate elliptic-parabolic-hyperbolic problem $$ b(u)_t - {\rm div} \tilde{\mathfrak a}(u,\nablaϕ(u))+ψ(u)=f, \quad u|_{t=0}=u_0 $$ in a bounded domain with homogeneous Dirichlet boundary conditions. The nonlinearities $b,ϕ$ and $ψ$ are supposed to be continuous non-decreasing, and the nonlinearity $\tilde{\mathfrak a}$ falls within the Leray-Lions framework. Some restrictions are imposed on the dependence of $\tilde{\mathfrak a}(u,\nablaϕ(u))$ on $u$ and also on the set where $ϕ$ degenerates. A model case is $\tilde{\mathfrak a}(u,\nablaϕ(u)) =\tilde{\mathfrak{f}}(b(u),ψ(u),ϕ(u))+k(u)\mathfrak{a}_0(\nablaϕ(u)),$ with $ϕ$ which is strictly increasing except on a locally finite number of segments, and $\mathfrak{a}_0$ which is of the Leray-Lions kind. We are interested 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 $b,ψ,ϕ,\tilde{\mathfrak{a}}$. Similar result is shown for the degenerate elliptic problem which corresponds to the case of $b\equiv 0$ and general non-decreasing surjective $ψ$. Existence, uniqueness and continuous dependence on data $u_0,f$ are shown when $[b+ψ](\R)=\R$ and $ϕ\circ [b+ψ]^{-1}$ is continuous.