Summary
Existence and regularity results for global solutions of the semilinear parabolic equation
with Dirichlet boundary conditions resp. of the heat equation with the boundary condition
are given, where «*»denotes convolution with respect to the t-variable. The main condition on g is a growth restriction;in the case of 3⩾nspace variables, the conditions coincide with the weakest known properties of g that guarantee the existence of global solutions of
Systems of equations of the structure (1)are also discussed.
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References
P. Aviles -J. Sandefur,Nonlinear Second Order Equations with Applications to Partial Differential Equations, J. Diff. Eq.,58 (1985), pp. 404–427.
V.Barbu,Nonlinear Volterra Integro-Differential Equations in Hilbert Spaces, Conf. Sem. Mat. Bari,143 (1976).
H. Engler,A Version of the Chain Rule and Integro-Differential Equations in Hilbert Spaces, SIAM J. Math. Anal.,13 (1982), pp. 801–810.
H.Engler,On Some Parabolic Integro-Differential Equations: Existence and Asymptotics of Solutions, in «Proceedings Equadiff '82» (Ed.: H. W.Knobloch, K.Schmitt), L.N. Math.,1017 (1983), pp. 161–167.
H. Engler,Stabilization of Solutions for a Class of Parabolic Integro-Differential Equations, Nonlinear Analysis: TMA,8 1984), pp. 1337–1371.
H.Engler,Bootstrapping for Higher Order Evolution Equations and Applications, in preparation.
L. C. Evans,Regularity Properties for the Heat Equation Subject to Nonlinear Boundary Constraints, Nonlinear Analysis: TMA,1 (1977), pp. 593–602.
L. C.Evans,Applications of Nonlinear Semigroup Theory to Certain Partial Differential Equations, Proceedings, Madison, WI (1977), pp. 163–188.
A.Friedman,Partial Differential Equations, New York, 1969.
G. Gripenberg,On some Integral and Integro-differential Equations in a Hilbert Space, Ann. Mat. Pura Appl.,118 (1978), pp. 181–194.
P. Grisvard -G. da Prato,Sommes d'Opérateurs Linéaires et Equations Différentielles Opérationelles, J. Math. Pures Appl.,54 (1975), pp. 305–387.
M. L. Heard,An Abstract Parabolic Volterra Integro-differential Equation, SIAM J. Math. Anal.,13 (1982), pp. 81–105.
H. Kielhöfer,Global Solutions of Semilinear Evolution Equations Satisfying an Energy Inequality, J. Diff. Eq.,36 (1980), pp. 188–222.
K. Kunisch -W. Schappacher,Order preserving Evolution Operators of Functional Differential Equations, Boll. Un. Mat. Ital., V,16-B (1979), pp. 480–500.
O. A.Ladyzenskaya - V. A.Solonnikov - N. N.Ural'tseva,Linear and Quasilinear Equations of Parabolic Type, Providence, R.I., 1968.
J. L. Lions -E. Magenes,Problemi ai limiti non omogenei, V, Ann. Scuola Norm. Sup. Pisa,16 (1962), pp. 1–44.
S.-O. Londen -J. A. Nohel,Nonlinear Volterra Integrodifferential Equations Occurring in Heat Flow, J. Integral Equations,6 (1984), pp. 11–50.
R. K.Miller,Nonlinear Volterra Integral Equations, Menlo Park, CA, 1971.
H. Pecher,Globale klassische Lösungen des Cauchyproblems semilinearer parabolischer Differentialgleichungen, J. Functional Analysis,20 (1975), pp. 286–303.
F. Rothe,Uniform Bounds from Bounded L p-Functionals in Reaction-Diffusion Equations, J. Diff. Eq.,45 (1982), pp. 207–233.
J. Sandefur,Existence and Uniqueness of Solutions of Second Order Nonlinear Differential Equations, SIAM J. Math. Anal.,14 (1983), pp. 477–487.
J. T.Schwartz,Nonlinear Functional Analysis, New York, 1969.
H. Ugowski,Existence of Solutions of Integro-Differential Equations of Parabolic Type, Ann. Polon. Math.,25 (1972), pp. 311–323.
W. v. Wahl,Starke Lösungen semilinearer parabolischer Differentialgleichungen, Math. Z.,144 (1975), pp. 257–263.
W. v. Wahl,Neue Resolventenabschätzungen für elliptische Differentialoperatoren und semilineare parabolische Gleichungen, Abh. Math. Sem. Hamburg,46 (1977), pp. 179–204.
G. F. Webb,An Abstract Semilinear Volterra Integrodifferential Equation, Proc. AMS,69 (1978), pp. 225–260.
N. D. Alikakos,Quantitative Maximum Principles and Strongly-coupled Gradient-like Reaction-diffusion Systems, Proc. Roy. Soc., Edinburgh, sect. A,94 (1983), pp. 265–286.
N. D. Alikakos,Regularity and Asymptotic Behavior for the Second Order Parabolic Equation with Nonlinear Boundary Conditions, J. Diff. Eq.,39 (1981), pp. 311–344.
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Engler, H. Existence of regular solutions for semilinear parabolic integro-differential equations. Annali di Matematica pura ed applicata 144, 23–47 (1986). https://doi.org/10.1007/BF01760809
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DOI: https://doi.org/10.1007/BF01760809