Dłotko, Tomasz Fourth order semilinear parabolic equations. (English) Zbl 0798.35078 Tsukuba J. Math. 16, No. 2, 389-405 (1992). The aim of this paper is to give a simple proof of the existence of a smooth solution to the semilinear parabolic equation with fourth order elliptic operator: \[ u_ t= -\varepsilon^ 2 \Delta^ 2 u+ f(t,x,u,u_ x,u_{xx}),\tag{1} \] \(x\in \Omega\subset \mathbb{R}^ n\), \(\Omega\) is a bounded domain, \(t\in [0,T_{\max})\), \(T_{\max}\leq +\infty\). We consider (1) together with initial-boundary conditions \[ u(0,x)= u_ 0(x),\quad x\in \Omega,\quad {\partial u\over\partial n}= {\partial(\Delta u)\over \partial n}=0\quad\text{when } x\in \partial\Omega. \] The general scheme of our proof of local existence is similar to the classical proof of the Picard theorem for solutions of ordinary differential equations. Cited in 1 ReviewCited in 1 Document MSC: 35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations 35K35 Initial-boundary value problems for higher-order parabolic equations Keywords:estimates of life-span; classical solvability; semilinear parabolic equation with fourth order elliptic operator PDFBibTeX XMLCite \textit{T. Dłotko}, Tsukuba J. Math. 16, No. 2, 389--405 (1992; Zbl 0798.35078) Full Text: DOI