On the existence of maximal and minimal solutions for parabolic partial differential equations

Bebernes, J. W.; Schmitt, K.

doi:10.1090/S0002-9939-1979-0516467-3

On the existence of maximal and minimal solutions for parabolic partial differential equations
HTML articles powered by AMS MathViewer

by J. W. Bebernes and K. Schmitt PDF

Proc. Amer. Math. Soc. 73 (1979), 211-218 Request permission

Abstract:

The existence of maximal and minimal solutions for initial-boundary value problems and the Cauchy initial value problem associated with $Lu = f(x,t,u,\nabla u)$ where L is a second order uniformly parabolic differential operator is obtained by constructing maximal and minimal solutions from all possible lower and all possible upper solutions, respectively. This approach allows f to be highly nonlinear, i.e., f locally Hölder continuous with almost quadratic growth in $|\nabla u|$.

References

Kiyoshi Akô, On the Dirichlet problem for quasi-linear elliptic differential equations of the second order, J. Math. Soc. Japan 13 (1961), 45–62. MR 147758, DOI 10.2969/jmsj/01310045
Herbert Amann, On the existence of positive solutions of nonlinear elliptic boundary value problems, Indiana Univ. Math. J. 21 (1971/72), 125–146. MR 296498, DOI 10.1512/iumj.1971.21.21012
J. W. Bebernes and K. Schmitt, Invariant sets and the Hukuhara-Kneser property for systems of parabolic partial differential equations, Rocky Mountain J. Math. 7 (1977), no. 3, 557–567. MR 600519, DOI 10.1216/RMJ-1977-7-3-557
H. Fujita and S. Watanabe, On the uniqueness and non-uniqueness of solutions of initial value problems for some quasi-linear parabolic equations, Comm. Pure Appl. Math. 21 (1968), 631–652. MR 234129, DOI 10.1002/cpa.3160210609
Herbert B. Keller, Elliptic boundary value problems suggested by nonlinear diffusion processes, Arch. Rational Mech. Anal. 35 (1969), 363–381. MR 255979, DOI 10.1007/BF00247683

Linear and quasilinear equations of parabolic type

W. Mlak, Parabolic differential inequalities and Chaplighin’s method, Ann. Polon. Math. 8 (1960), 139–153. MR 116135, DOI 10.4064/ap-8-2-139-153
W. Mlak, An example of the equation $u_{t}=u_{xx}+f(x, t,u)$ with distinct maximum and minimum solutions of a mixed problem, Ann. Polon. Math. 13 (1963), 101–103. MR 149119, DOI 10.4064/ap-13-1-101-103
Mitio Nagumo, On principally linear elliptic differential equations of the second order, Osaka Math. J. 6 (1954), 207–229. MR 70014
C. V. Pao, Successive approximations of some nonlinear initial-boundary value problems, SIAM J. Math. Anal. 5 (1974), 91–102. MR 339507, DOI 10.1137/0505010
C. V. Pao, Positive solutions of a nonlinear boundary-value problem of parabolic type, J. Differential Equations 22 (1976), no. 1, 145–163. MR 422876, DOI 10.1016/0022-0396(76)90008-5
Giovanni Prodi, Teoremi di esistenza per equazioni alle derivate parziali non lineari di tipo parabolico. I, II, Ist. Lombardo Sci. Lett. Rend. Cl. Sci. Mat. Nat. (3) 17(86) (1953), 3–26, 27–47 (Italian). MR 0064287
J.-P. Puel, Existence, comportement à l’infini et stabilité dans certains problèmes quasilinéaires elliptiques et paraboliques d’ordre $2$, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 3 (1976), no. 1, 89–119. MR 399654
Ray Redheffer and Wolfgang Walter, Counterexamples for parabolic differential equations, Math. Z. 153 (1977), no. 3, 229–236. MR 433036, DOI 10.1007/BF01214476
D. H. Sattinger, Monotone methods in nonlinear elliptic and parabolic boundary value problems, Indiana Univ. Math. J. 21 (1971/72), 979–1000. MR 299921, DOI 10.1512/iumj.1972.21.21079
Klaus Schmitt, Boundary value problems for quasilinear second-order elliptic equations, Nonlinear Anal. 2 (1978), no. 3, 263–309. MR 512661, DOI 10.1016/0362-546X(78)90019-6
Friedrich Tomi, Über semilineare elliptische Differentialgleichungen zweiter Ordnung, Math. Z. 111 (1969), 350–366 (German). MR 279428, DOI 10.1007/BF01110746