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On parabolic systems with variational structure. (English) Zbl 0593.35054
The author deals with the parabolic system of the second order for a vector function \(u=(u^ 1,...,u^ N)\), \(N>1\), \[ u^ i_ t=\sum_{\alpha,\beta}D_{\alpha}(a_{\alpha,\beta}(x,u)D_{\beta}u^ i)+a^ i_ 0(x,u\quad,\nabla u),\quad i=1,...,N, \] in the cylinder \(Q=\Omega \times (0,T)\), where \(\Omega \subset R^ n\) is a bounded Lipschitz domain. The functions \(a_{\alpha,\beta}\), \(a^ i_ 0\) are Lipschitz, \(a_{\alpha,\beta}\)-bounded, \(a^ i_ 0(x,u,p)\) having quadratic growth in p. Uniform ellipticity in second order term and the variational structure of the system is assumed. Moreover a one-sided condition is imposed: \(a_ 0(x,u(x),\nabla u(x)):\) \(u\geq -\mu | \nabla u(x)|^ 2-K\), for each \(u\in L_{\infty}(\Omega)\cap W^ 1_ 2(\Omega)\) and a.e. \(x\in \Omega\) \((\mu,K>0)\). Employing the method of continuation the existence of a solution is considered for the Cauchy- Dirichlet problem \[ u(x,0)=u_ 0(x),\quad x\in \Omega;\quad u(x,t)=g(x),\quad x\in \partial \Omega, \] with g small in \(L_{\infty}(\partial \Omega)\)-norm. The main result yields (under some regularity conditions on \(u_ 0,g)\) a solution-Hölder continuous in \(\bar Q,\) for the case \(n=2\). For general \(n\geq 2\) the problem is reduced to a certain condition which is proved to hold for \(n=2\).
Reviewer: P.Polacik

MSC:
35K55 Nonlinear parabolic equations
35K45 Initial value problems for second-order parabolic systems
35K50 Systems of parabolic equations, boundary value problems (MSC2000)
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
35J85 Unilateral problems; variational inequalities (elliptic type) (MSC2000)
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References:
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