Kuttler, Kenneth; Aifantis, Elias Quasilinear evolution equations in nonclassical diffusion. (English) Zbl 0675.35053 SIAM J. Math. Anal. 19, No. 1, 110-120 (1988). 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 Cited in 30 Documents MSC: 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) Keywords:viscosity; higher order; gradient effects; existence; quasilinear evolution equation; reflexive Banach spaces; diffusion models PDFBibTeX XMLCite \textit{K. Kuttler} and \textit{E. Aifantis}, SIAM J. Math. Anal. 19, No. 1, 110--120 (1988; Zbl 0675.35053) Full Text: DOI Link