A parabolic hemivariational inequality. (English) Zbl 0858.35072

A quasilinear nonmonotone parabolic initial boundary value problem of the form \[ u'(t)+Au(t)+\Sigma(t)=f(t), \quad u(0)=u_0, \quad \Sigma(x,t)\in\widehat{b} (u(x,t)) \quad\text{a.e. }(x,t)\in Q_T, \] is considered taking into account the existence of its solutions. This is a generalization of stationary hemivariational inequalities to the dynamic case in which \(A\) is assumed to be linear and continuous, and the multivalued mapping \(\widehat{b}\) is constructed by filling the gaps of a function \(b\in L^\infty_{\text{loc}}(R)\). In the nonpotential case, \(A\) being nonsymmetric, a linear growth condition on the behavior of \(b\) at infinity has been imposed whereas in the potential case, \(A\) being symmetric, a sign condition on \(b\) has been introduced. The Galerkin method in the setting of the evolution triple “\(V\subset H\subset V' \)” with \(V\hookrightarrow H^1(\Omega)\) and \(H=L^2(\Omega)\), combined with compactness technique are applied as the main mathematical tools.


35K85 Unilateral problems for linear parabolic equations and variational inequalities with linear parabolic operators
35R70 PDEs with multivalued right-hand sides
49J40 Variational inequalities
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[1] Browder, F. E.; HESS, P., Nonlinear mappings of monotone type in Banach spaces,, J. fund. Analysis, 11, 251-294 (1972) · Zbl 0249.47044
[2] Carl, S.; Heikkilä, S., An existence result for elliptic differential inclusions with discontinuous nonlinearity,, Nonlinear Analysis, 18, 471-479 (1992) · Zbl 0755.35039
[4] Chang, K. C., Variational methods for nondifferentiable functional and their applications to partial differential equations,, J. math. Analysis Applic., 80, 102-129 (1981)
[5] Costa, D. G.; Goncalves, J. V.A., Critical point theory for nondifferentiable functionals and applications,, J. math. Analysis Applic., 153, 470-485 (1990) · Zbl 0717.49007
[6] Heikkilä, S.; Lakshmikantham, V., (Monotone Iterative Techniques for Discontinuous Nonlinear Differential Equations (1994), Marcel Dekker: Marcel Dekker Providence, R.I.) · Zbl 0804.34001
[7] Miettinen, M.; Haslinger, J., Approximation of optimal control problems of hemivariational inequalities,, Num. funct. Analysis Optim., 13, 43-68 (1992) · Zbl 0765.49007
[8] Miettinen, M., Approximation of hemivariational inequalities and optimal control problems, (Report 59. Report 59, PhD Thesis (1993), University of Jyväskylä, Department of Mathematics: University of Jyväskylä, Department of Mathematics New York) · Zbl 0790.49012
[9] Naniewicz, Z., On the pseudo-monotonicity of generalized gradients of nonconvex functions,, Applic. Analysis, 47, 151-172 (1992) · Zbl 0724.49011
[10] Panagiotopoulos, P. D., Nonconvex superpotentials in sense of F. H. Clarke and applications,, Mech. Res. Comm., 8, 335-340 (1981) · Zbl 0497.73020
[11] Panagiotopoulos, P. D., Inequality Problems in Mechanics and Applications. Convex and Nonconvex Energy Functions (1985), Birkhäuser · Zbl 0579.73014
[12] Panagiotopoulos, P. D., Hemivariational inequalities and their applications, (Moreau, J. J.; Panagiotopoulos, P. D.; Strang, G., Topics in Nonsmooth Mechanics (1988), Birkhäuser: Birkhäuser Boston) · Zbl 0655.73010
[13] Panagiotopoulos, P. D., Coercive and semicoercive hemivariational inequalities,, Nonlinear Analysis, 16, 209-231 (1991) · Zbl 0733.49012
[14] Panagiotopoulos, P. D., (Hemivariational Inequalities and Applications in Mechanics and Engineering (1993), Springer: Springer Boston) · Zbl 0826.73002
[15] Rauch, J., Discontinuous semilinear differential equations and multiple valued maps,, (Proc. Am. math. Soc., 64 (1977)), 277-282 · Zbl 0413.35031
[16] Carl, S.; Heikkilä, S., Extremal solutions of quasilinear parabolic boundary value problems with discontinmuous nonlinearities,, Dyn. Syst. Applic., 3, 251-258 (1994) · Zbl 0803.35077
[17] Frigon, M.; Saccon, C., Heat equations with discontinuous nonlinearities on convex and nonconvex constraints,, Nonlinear Analysis, 17, 923-946 (1991) · Zbl 0753.35046
[18] Shuzhong, S., Nagumo type condition for partial differential inclusions,, Nonlinear Analysis, 12, 951-967 (1988) · Zbl 0654.49016
[19] Landes, R.; Mustonen, V., A strongly nonlinear parabolic initial boundary value problem,, Ark. f. Mat., 25, 29-40 (1987) · Zbl 0697.35071
[20] Lions, J.-L., (Quelques méthodes de résolution des problémes aux limites non linéaires (1969), Dunod: Dunod New York) · Zbl 0189.40603
[21] Brezis, H., Collection mathématiques pour la maitrise, (Analyse fonctionelle. Théorie et applications (1987), Masson: Masson Paris)
[22] Landes, R., On the existence of weak solutions for quasilinear parabolic initial-boundary value problems,, (Proc. R. Soc. Edinb., A89 (1981)), 217-237 · Zbl 0493.35054
[23] Zeidler, E., (Nonlinear Functional Analysis and its Applications II/A (1990), Springer: Springer Paris) · Zbl 0462.47045
[24] Landes, R., A note on strongly nonlinear parabolic equations of higher order,, Diff. Integral Eqns, 3, 851-862 (1990) · Zbl 0733.35052
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