On the Existence of Maximal and Minimal Solutions for Parabolic Partial Differential Equations

Abstract

Abstract. The existence of maximal and minimal solutions for initialboundary value problems and the Cauchy initial value problem associated with Lu -f(x, t, u, Vu) 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 / to be highly nonlinear, i.e., / locally Holder continuous with almost quadratic growth in |V«|.

Maximal and minimal solutions to a class of elliptic quasilinear problems

“…with 7 e]0,1[ and L > 0 dependent only on o. K. ||<7||z,vm)< IMIl^q)-Thus the "second step" in the exist once theory for a wide class of quasilinear elliptic equations of the general form (3) a'J(x, u. ¡f.r)«.r,.t'i I d(.r.…”

Semilinear Periodic-Parabolic Equations with Nonlinear Boundary Conditions

“…(7) has at least one classical solution u # C 1++, (1++)Â2 (0 _I) such that inequality (36) holds. Since u # C 1, 1Â2 (0 _I ), by using the fact that the function h is locally Lipschitz continuous as defined in (6), one deduces that h( }, }, u( } , } )) # C +, +Â2 (0 _I ). Moreover, since…”

“…Let h: 0_I_R Ä R be a locally Lipschitz continuous function which is T-periodic in t; that is, h(x, 0, u)=h(x, T, u) for all (x, u) # 0_R, and for each u # R there exist a (closed) interval U/R about u and a number M>0 such that |h(x, t, v)&h( y, s, w)| M(|x&y| 2 +|t&s|+ |v&w| 2 ) 1Â2 (6) for all (x, t, v), ( y, s, w) # 0_I_U. (Note that, strictly speaking, the function h(x, }, u) is (1Â2)-Ho lder continuous in the variable t, uniformly for (x, u) # 0_U.)…”

“…Since u # W 2, 1 p (0_I )/C 1++, (1++)Â2 (0 _I) and the function h is (locally) Lipschitz in the sense defined in (6), it follows that the function defined by…”

Maximal and Minimal Solutions to a Class of Elliptic Quasilinear Problems

“…For any r E [0,1] and any v E C71(fi) we consider the problem: find u E W2(Q) such that To prove this theorem we adapt Akô's method (cf. [1,3]). We define «max(i) = sup{u(x) | u solves (5), (6)}, umin(i) = inf{ii(x) I u solves (5), (6)}, and proceed to prove that both ttmax and wmin are solutions.…”