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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

t(z-Δz)- i (D(z) i z)+Δ 2 z=h,

z(t,·)=w(t,·) on Ω, Δz(t,·)=k(t,·) on Ω,

lim t0 + Ω (z(t)-u 0 )v+(z(t)-u 0 )·vdx=0,

vV={uH 2 (Ω): u(x)=0 on Ω}, where u 0 is the initial condition.

Reviewer: G.F.Webb
MSC:
35K55Nonlinear parabolic equations
35G10Initial value problems for linear higher-order PDE
35K25Higher order parabolic equations, general
34G20Nonlinear ODE in abstract spaces
35A05General existence and uniqueness theorems (PDE) (MSC2000)