论文标题
完全非线性进化的良好的二阶包括
Well-posedness of a fully nonlinear evolution inclusion of second order
论文作者
论文摘要
二阶非线性进化的包含方程的抽象\ textsc {cauchy}问题的拟定性\ t> 0，\\ u（0）= u_0，\ quad u'（0）= v_0 \ end {cases} \ end end {align*}在真正的可分离\ textsc {hilbert} space $ \ mathscr {hilbert} space $ \ mathscr {h h} $中ψ）} \ cap d（ψ），f \ in l^2（0，t; \ mathscr {h}）$。功能性$ψ：\ Mathscr {h} \ rightarrow（ - \ infty，+\ infty] $应该是适当的，较低的半连续性和convex和convex和nonlinear operator $ b：[0，t] \ times \ times \ times \ mathscr {h}条件。
The well-posedness of the abstract \textsc{Cauchy} problem for the doubly nonlinear evolution inclusion equation of second order \begin{align*} \begin{cases} u''(t)+\partial Ψ(u'(t))+B(t,u(t))\ni f(t), &\quad t\in (0,T),\, T>0,\\ u(0)=u_0, \quad u'(0)=v_0 \end{cases} \end{align*} in a real separable \textsc{Hilbert} space $\mathscr{H}$, where $u_0\in \mathscr{H}, v_0\in \overline{D(\partial Ψ)}\cap D(Ψ), f\in L^2(0,T;\mathscr{H})$. The functional $Ψ: \mathscr{H} \rightarrow (-\infty,+\infty]$ is supposed to be proper, lower semicontinuous, and convex and the nonlinear operator $B:[0,T]\times \mathscr{H}\rightarrow \mathscr{H}$ is supposed to satisfy a (local) \textsc{Lipschitz} condition. Existence and uniqueness of strong solutions $u\in H^2(0,T^*;\mathscr{H})$ as well as the continuous dependence of solutions from the data re shown by employing the theory of nonlinear semigroups and the Banach fixed-point theorem. If $B$ satisfies a local Lipschitz condition, then the existence of strong local solutions are obtained.
