×

zbMATH — the first resource for mathematics

Existence theorems for semilinear parabolic systems with functionals. (English) Zbl 0543.35052
On considère des systèmes d’équations différentielles partielles semi-linéaires de la forme \(L(u)=f(t,x,u(\cdot)),\) avec des conditions au bord soit \(u_ k=0\) soit \(\partial u_ k/\partial \nu_ k=g_ k(t,x,u(\cdot))\) ou \(u=(u_ 1,u_ 2,...,u_ N).\) L’opérateur L est linéaire uniformément parabolique. L’auteur met en évidence de bonnes hypothèses sur les non-linéarités pour obtenir, grâce à des potentiels bien choisis et à des théorèmes de point fixe, des théorèmes d’existence de solutions des problèmes posés, en un sens faible et fort.
Reviewer: M.-T.Lacroix

MSC:
35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
35D05 Existence of generalized solutions of PDE (MSC2000)
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Amann, H., Invariant sets and existence theorems for semilinear parabolic and elliptic systems, J. math. analysis applic., 65, 432-467, (1978) · Zbl 0387.35038
[2] Brzychczy St., Approximate iterative method and existence of solutions of non-linear parabolic differential- functional equations. To appear in Ann. Polon. Math. · Zbl 0537.35044
[3] Cushing, J.M., Integrodifferential equations and delay models in population dynamics, () · Zbl 0363.92014
[4] Fitzgibbon, W.E., Semilinear functional differential equations in Banach space, J. diff. eqns, 29, 1-14, (1978) · Zbl 0392.34041
[5] Friedman, A., Partial differential equations of parabolic type, (1964), Prentice-Hall Englewood-Cliffs · Zbl 0144.34903
[6] Il’in, A.M.; Kalashnikov, A.S.; Oleinik, O.A., Linear equations of the second order of parabolic type, Russ. math. surveys, 17, 1-143, (1962) · Zbl 0988.35001
[7] Kuiper, H.J., Existence and comparison theorems for nonlinear diffusion systems, J. analysis applic., 60, 166-181, (1977) · Zbl 0358.35043
[8] Kuiper, H.J., Invariant sets for nonlinear elliptic and parabolic systems, SIAM J. math. analysis, 11, 1075-1103, (1980) · Zbl 0468.35055
[9] Ladyzhenskaja, O.A.; Solonnikov, V.A.; Uraltseva, N.N., Linear and quasilinear equations of parabolic type, ()
[10] Marcati, P.; Pozio, M.A., Global asymptotic stability for a vector disease model with spatial spread, J. math. biol., 9, 179-187, (1980) · Zbl 0419.92013
[11] Pogorzelski, W., Propriétés des intégrales de l’équation parabolique normale, Ann. polon. math., 4, 61-92, (1957/1958) · Zbl 0080.30602
[12] Pogorzelski, W., Premier problème de Fourier pour l’équation parabolique dont LES coefficients dépendent de la fonction inconnue, Ann. polon. math., 6, 15-40, (1959) · Zbl 0085.31002
[13] Politjukov, V.P.; Politjukov, V.P., On the theory of upper and lower solutions and the solvability of quasilinear integro-differential equations, Math. USSR-sb., Math. USSR-sb., 35, 499-507, (1979) · Zbl 0431.45010
[14] Rankin, S.M., Existence and asymptotic behavior of a functional differential equation in Banach space, J. math. analysis applic., 88, 531-542, (1982) · Zbl 0519.34045
[15] Redheffer R., The dependence of quenching on memory, unpublished manuscript.
[16] Redheffer R, & Walter W., Uniqueness, stability and error estimation for parabolic functional-differential equations. Published in the Jubilee Volume dedicated to the 70th anniversary of Academician I. N. Vekua. · Zbl 0425.35053
[17] Redheffer, R.; Walter, W., Stability of the null solution of parabolic functional inequalities, Trans. am. math. soc., 262, 285-302, (1980) · Zbl 0454.35015
[18] Triggiani, R., Well-posedness and regularity of boundary feedback parabolic systems, J. diff. eqns, 36, 347-362, (1980) · Zbl 0438.35042
[19] Ugowski, H., Some theorems on the estimate and existence of solutions of integro-differential equations of parabolic type, Ann. polon. math., 25, 311-323, (1971) · Zbl 0229.45009
[20] Ugowski, H., On a certain non-linear initial-boundary value problem for integro-differential equations of parabolic type, Ann. polon. math., 28, 249-259, (1973) · Zbl 0243.45014
[21] Walter, W., Differential and integral inequalities, ()
[22] Redlinger, R., Existenzsätze für semilineare parabolische systeme mit funktionalen, () · Zbl 0535.35038
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.