Global existence and blow-up analysis for some degenerate and quasilinear parabolic systems

Haihua Lu

doi:10.14232/ejqtde.2009.1.49

Abstract

This paper deals with positive solutions of some degenerate and quasilinear parabolic systems not in divergence form: u1t = f1(u2)(�u1 + a1u1), � � � ,u(n 1)t = fn 1(un)(�un 1 + an 1un 1), unt = fn(u1)(�un+anun) with homogenous Dirichlet boundary condition and posi- tive initial condition, where ai (i = 1,2, �� ,n) are positive constants and fi (i = 1,2, �� ,n) satisfy some conditions. The local existence and uniqueness of classical solution are proved. Moreover, it will be proved that: (i) when min{a1, � � � , an} ≤ �1 then there exists global positive classical solution, and all positive classical solutions can not blow up in finite time in the meaning of maximum norm; (ii) when min{a1, � � � , an} > �1, and the initial datum (u10, � � � , un0) satisfies some assumptions, then the positive classical solution is unique and blows up in finite time, where �1 is the first eigenvalue of −� in with homogeneous Dirichlet boundary condition.

Highlights

Introduction and main resultIn this paper, we consider the following degenerate and quasilinear parabolic systems not in divergence form: uit = fi(ui+1)(∆ui +aiui), x ∈ Ω, t > 0, i = 1, 2, · · ·, n − 1, unt fn(u1)(∆un anun), x ∈ Ω, t > 0, ui(x, 0) ui0(x), ui(x, t)0, x ∈ Ω, i = 1, 2, · · ·, n, x ∈ ∂Ω, t > 0, i = 1, 2, · · ·, n, (1.1)where ai are positive constants, fi, ui0(x), i = 1, 2, · · ·, n, satisfy (H1) ui0(x) ∈ C1(Ω ), ui0(x) > 0 in Ω, ; (H2) ui0(x) and
Our present results develop the work of [12]
Let ε < 1, (u1ε, · · ·, unε) be the solution of (2.3), for any fixed T : 0 < T < min{T (ε), T ∗}, uiε(x, t) ≤ gi(t), ∀ (x, t) ∈ Ω × [0, T ], i = 1, 2, · · ·, n, which implies that T (ε) ≥ T ∗ for all ε < 1