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arXiv 0810.2326math.AP
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三重非线性degenerate parabolic equations的适定性结果

Well-posedness results for triply nonlinear degenerate parabolic equations

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

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摘要：

研究了具有齐次Dirichlet边界条件的三重非线性退化椭圆-抛物-双曲问题$$ b(u)_t - {\rm div} \tilde{\mathfrak a}(u,\nablaϕ(u))+ψ(u)=f, \quad u|_{t=0}=u_0 $$在有界区域上的适定性.假设非线性度$b,ϕ$和$ψ$是连续非减的，并且非线性度$\tilde{\mathfrak a}$落入Leray-Lions框架内。对$\tilde{\mathfrak a}(u,\nablaϕ(u))$对$u$的依赖性以及$ϕ$退化的集合施加了一些限制。一个模型的情况是$\tilde{\mathfrak a}(u,\nablaϕ(u)) =\tilde{\mathfrak{f}}(b(u),ψ(u),ϕ(u))+k(u)\mathfrak{a}_0(\nablaϕ(u)),$与$ϕ$是严格增加的，除了在一个局部有限数量的段，和$\mathfrak{a}_0$这是勒雷-狮子类。我们感兴趣的是熵解的存在性、唯一性和稳定性。如果$b=\mathrm{Id}$，我们得到一个一般的连续依赖的结果，数据$u_0,f$和非线性$b,ψ,ϕ,\tilde{\mathfrak{a}}$。对于退化椭圆型问题，对应于$b\equiv 0$和一般非减满射$ψ$的情形，也得到了类似的结果。当$[b+ψ](\R)=\R$和$ϕ\circ [b+ψ]^{-1}$连续时，证明了解的存在性、唯一性和对数据$u_0,f$的连续依赖性.

英文摘要：

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.

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