# zbMATH — the first resource for mathematics

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)
Full Text:
##### References:
  ARONSON, D.G.: Non-negative solutions of linear parabolic equations; Ann. Sc. Norm. Sup. Pisa,22(1968), 607-694 · Zbl 0182.13802  EELLS, J., J.H. SAMPSON: Harmonic mappings of Riemannian manifolds; Am. J. Math.86(1964), 109-160 · Zbl 0122.40102  FREHSE, J.: On Two-Dimensional Quasi-Linear Elliptic Systems, manusc. math.28(1979), 21-49 · Zbl 0415.35025  GIAQUINTA, M., M. STRUWE: An optimal regularity result for a class of quasilinear parabolic systems, manusc. math.36(1981), 223-239 · Zbl 0475.35026  GILBARG, D., N.S. TRUDINGER: Elliptic Partial Differential Equations of Second Order, Springer Grundlehren 224, Berlin, Heidelberg, New York, Tokyo, 2nd edition, 1983 · Zbl 0562.35001  HAMILTON, R.S.: Harmonic Maps of Manifolds with Boundary, Springer Lecture Notes 471, Berlin, Heidelberg, New York, 1975 · Zbl 0308.35003  HILDEBRANDT, S., H. KAUL, K.-O. WIDMAN: An Existence Theorem for Harmonic Mappings of Riemannian Manifolds, Acta Math.138 (1977), 1-16 · Zbl 0356.53015  JOST, J.: Ein Existenzbeweis für harmonische Abbildungen, die ein Dirichletproblem lösen, mittels der Methode des Wärmeflusses; manusc. math.34(1981), 17-25 · Zbl 0459.58013  LADYSHENSKAYA, O.A., V.A. SOLONNIKOV, N.N. URAL’CEVA: Linear and Quasilinear Equations of Parabolic Type; Transl. Math. Monogr. 23, AMS, Providence, 1968  LIONS, J.L.: Quelques méthodes de résolution des problèmes aux limites non-linéaires, Dunod, Paris, 1969  MORREY, C.B. Jr.: Multiple Integrals in the Calculus of Variations, Springer, New York, 1966 · Zbl 0142.38701  STRUWE, M.: On the Hölder continuity of bounded weak solutions of quasilinear parabolic systems; manusc. math.35(1981), 125-145 · Zbl 0519.35007  STRUWE, M.: Some regularity results for quasilinear parabolic systems, to appear in: Commentat. Math. Univ. Carol.  STRUWE, M.: On the evolution of harmonic mappings, to appear in: Comm. Math. Helv. · Zbl 0595.58013  TOLKSDORF, P.: On some parabolic variational problems with quadratic growth, to appear in: Ann. Sc. Norm. Sup. Pisa  WAHL, W. VON: Verhalten der Lösungen parabolischer Gleichungen für t?? mit Lösbarkeit im Großen; Nachr. Akad. Wiss. Göttingen5(1981) · Zbl 0492.35035  WIEGNER, M.: On two-dimensional elliptic systems with a one-sided condition, Math. Z.178(1981), 493-500 · Zbl 0474.35046
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.