Boundedness and blowup for nonlinear degenerate parabolic equations

Abstract

The author deals with the quasilinear parabolic equation In this paper we consider the following quasilinear parabolic equation:Email address: george chen@cbu.ca (Shaohua Chen).

Large time behavior of solutions to degenerate parabolic equations

“…r ≥ p + 1 and b < 0, then (3) holds. In fact, by [1] and [11], the positive classical solution exists for all t > 0, ε 0 ≤ u(x, t)/ψ(x) ≤ M , and by [5], the corresponding steady state exists. Under Condition I and II, (3) can be written as…”

“…In fact, by [1], the positive classical solution exists and u(x, t)/ ψ(x) ≤ M for some M > 0 and all t > 0. It is easy to see that the corresponding steady state exists.…”

“…The global existence of the classical solutions has been proved by many authors under various conditions on f and g, see [1][2][3][4]8,9,11]. Winkler [9] considered the following problem: where λ 1 is the first eigenvalue of −Δ with zero Dirichlet boundary condition.…”

Global existence and nonexistence for some degenerate and quasilinear parabolic systems

“…Mu et al [12] considered the same problems and obtained similar results. Chen [3] considered the system (1.2) with lower order terms f (u, v, Du) and g (u, v, Dv) and showed that all solutions are bounded if (1 + c 1 ) √ ab < λ 1 and blow up in a finite time if (1 + c 1 ) √ ab > λ 1 , where c 1 > −1 related to f and g. Chen [2] and Chen and Yu [4] also discussed single equations with lower order terms. Li et al [9] investigated the following strong coupled system u t = v p ( u + au) and v t = u q ( v + bv), (1.4) and proved that all solutions are global iff λ 1 min{a, b}.…”

“…In this method, we estimate the integral of a ratio of one solution to the other. This method shows successful in proving existence and blowup problems (see [1][2][3][4]). Then we use the method introduced by Li et al [9] to obtain a classical solution to (1.1).…”