Quasilinear evolution equations in nonclassical diffusion. (English) Zbl 0675.35053

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 \[ \frac{\partial}{\partial t}(z-\Delta z)- \partial_ i(D(z)\partial_ iz)+\Delta^ 2z=h, \] z(t,\(\cdot)=w(t,\cdot)\) on \(\partial \Omega\), \(\Delta z(t,\cdot)=k(t,\cdot)\) on \(\partial \Omega\), \[ \lim_{t\to 0^+} \int_{\Omega}(z(t)-u_ 0)v+\nabla (z(t)-u_ 0)\cdot \nabla vdx=0, \] \(v\in V=\{u\in H^ 2(\Omega):\) \(u(x)=0\) on \(\partial \Omega \}\), where \(u_ 0\) is the initial condition.
Reviewer: G.F.Webb


35K55 Nonlinear parabolic equations
35G10 Initial value problems for linear higher-order PDEs
35K25 Higher-order parabolic equations
34G20 Nonlinear differential equations in abstract spaces
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
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