Global solutions of abstract semilinear parabolic equations with memory terms

Abstract.The main purpose of this paper is to obtain the existence of global solutions to semilinear integro-differential equations in Hilbert spaces for rather general convolution kernels and nonlinear terms with superlinear growth at infinity. The included application to a nonlinear model of heat flow in materials of fading memory type provides motivations for the abstract theory.

[1]  Jan Prüss,et al.  Global existence for a semilinear parabolic Volterra equation , 1992 .

[2]  Enrique Fernández-Cara,et al.  Null controllability of the semilinear heat equation , 1997 .

[3]  Jace W. Nunziato,et al.  On heat conduction in materials with memory , 1971 .

[4]  乔花玲,et al.  关于Semigroups of Linear Operators and Applications to Partial Differential Equations的两个注解 , 2003 .

[5]  H. Beckert,et al.  J. L. Lions and E. Magenes, Non‐Homogeneous Boundary Value Problems and Applications, II. (Die Grundlehren d. Math. Wissenschaften, Bd. 182). XI + 242 S. Berlin/Heidelberg/New York 1972. Springer‐Verlag. Preis geb. DM 58,— , 1973 .

[6]  G. M.,et al.  Partial Differential Equations I , 2023, Applied Mathematical Sciences.

[7]  J. Nohel,et al.  Asymptotic Behavior of Solutions of Nonlinear Volterra Equations with Completely Positive Kernels , 1981 .

[8]  M. Gurtin,et al.  A general theory of heat conduction with finite wave speeds , 1968 .

[9]  Well-posedness of Semilinear Heat Equations with Iterated Logarithms , 1999 .

[10]  Vittorino Pata,et al.  Asymptotic behavior of a semilinear problem in heat conduction with memory , 1998 .

[11]  A. Haraux,et al.  Équations d'évolution avec non linéarité logarithmique , 1980 .

[12]  Jiongmin Yong,et al.  Optimal Control Theory for Infinite Dimensional Systems , 1994 .

[13]  J. Prüss.  Evolutionary Integral Equations And Applications , 1993 .

[14]  J. Lions,et al.  Non-homogeneous boundary value problems and applications , 1972 .

[15]  D. Sforza.  Existence in the Large for a Semilinear Integrodifferential Equation with Infinite Delay , 1995 .

[16]  A. Lunardi.  Analytic Semigroups and Optimal Regularity in Parabolic Problems , 2003 .

[17]  Olof J. Staffans,et al.  Volterra Integral and Functional Equations: Index , 1990 .

[18]  A. Lunardi,et al.  Fully nonlinear integrodifferential equations in general Banach space , 1985 .

[19]  Claudio Giorgi,et al.  Thermodynamic properties and stability for the heat flux equation with linear memory , 1993 .

[20]  P. Cannarsa,et al.  Well posedness and control of semilinear wave equations with iterated logarithms , 1999 .

[21]  Sebastian Anita,et al.  NULL CONTROLLABILITY OF NONLINEAR CONVECTIVE HEAT EQUATIONS , 2000 .

[22]  Vilmos Komornik,et al.  Controllability of semilinear wave equations with infinitely iterated logarithms , 2002 .

[23]  Daniel B. Henry.  Geometric Theory of Semilinear Parabolic Equations , 1989 .

[24]  Hector O. Fattorini,et al.  Infinite Dimensional Optimization and Control Theory: References , 1999 .

[25]  J. Meixner.  On the linear theory of heat conduction , 1970 .

[26]  S. Aizicovici.  On a semilinear volterra integrodifferential equation , 1980 .