A Singular Quasilinear Diffusion Equation in $l^{1}$
نویسندگان
چکیده
$+\beta(u)\ni f$ has a unique generalized or integral solution in $L^{1}$ when $\alpha$ and $\beta$ are maximal monotone graphs in $R$, each containing the origin, and at each point of their common domain either one of $\alpha$ or $\beta$ is single-valued. Weak Maximum and Comparison Principles follow from an $L^{\infty}$ estimate on the solution and from an $L^{1}$ estimate on the difference of solutions, respectively. This $L^{1}$ integral solution is shown to satisfy the above partial differential equation in the sense of distributions when $\alpha$ is surjective (or the data is bounded) and $\beta$ is continuous. We shall consider the initial-boundary-value problem (1.a) $u_{t}-\Delta v+w=f,$ $v\in\alpha(u),$ $w\in\beta(u)$ in $\Omega$ (1.b) $v=0$ on $\partial G\cross(O, T)$ (1.c) $u=u_{0}$ on $G\cross\{0\}$ where $G$ is a bounded domain in $R^{n},$ $\Omega\equiv G\cross(O, T),$ $\Delta$ is the Laplacian in $R^{n}$ , and $\alpha$ and $\beta$ are maximal monotone graphs in $R\cross R$, each containing the origin. The problem (1) will be regarded as an abstract Cauchy problem of the form (2.a) $u’(t)+A(u(t))+B(u(t))\ni f(t)$ , $a.e$ . $t\in(O, T)$ (2. b) $u(0)=u_{0}$ in the Banach space $L^{1}(G)$ . An integral solution of (2) in a Banach space $X$ is a $u\in C(O, T_{j}X)$ such that $u(O)=u_{0}$ and $u(t)\in dom(A+B)$ ,
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تاریخ انتشار 2008