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Zbl 0675.35053
Kuttler, Kenneth; Aifantis, Elias
Quasilinear evolution equations in nonclassical diffusion. (English)
[J] SIAM J. Math. Anal. 19, No.1, 110-120 (1988). ISSN 0036-1410; ISSN 1095-7154/e

The authors establish the existence of solutions to the abstract quasilinear evolution equation $(Bu)'+A(u)u=f(u)$ in reflexive Banach spaces. The abstract results are applied to various diffusion models. One such model is the equation $$ \frac{\partial}{\partial t}(z-\Delta z)- \partial\sb i(D(z)\partial\sb iz)+\Delta\sp 2z=h, $$ z(t,$\cdot)=w(t,\cdot)$ on $\partial \Omega$, $\Delta z(t,\cdot)=k(t,\cdot)$ on $\partial \Omega$, $$ \lim\sb{t\to 0\sp+} \int\sb{\Omega}(z(t)-u\sb 0)v+\nabla (z(t)-u\sb 0)\cdot \nabla vdx=0, $$ $v\in V=\{u\in H\sp 2(\Omega):$ $u(x)=0$ on $\partial \Omega \}$, where $u\sb 0$ is the initial condition.
[G.F.Webb]

MSC 2000:: *35K55 Nonlinear parabolic equations
35G10 Initial value problems for linear higher-order PDE
35K25 Higher order parabolic equations, general
34G20 Nonlinear ODE in abstract spaces
35A05 General existence and uniqueness theorems (PDE)

Keywords: viscosity; higher order; gradient effects; existence; quasilinear evolution equation; reflexive Banach spaces; diffusion models

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