Life span of solutions with large initial data for a semilinear parabolic system coupling exponential reaction terms

Sen Zhou

doi:10.1186/s13661-019-1269-y

Abstract

In this paper, we study a coupled systems of parabolic equations subject to large initial data. By using comparison principle and Kaplan’s method, we get the upper and lower bound for the life span of the solutions.

Highlights

In this paper, we consider the following nonlinear parabolic system: ⎧ ⎪⎪⎪⎪⎪⎨vutt = =u + emu+pv, v + equ+nv,⎪⎪⎪⎪⎪⎩uu((xx, t) = v(x, t) = 0) = λφ(x), 0, v(x, 0) = λψ(x), x ∈ Ω, t > 0, x ∈ Ω, t > 0, x ∈ ∂Ω, t > 0, x ∈ Ω, (1.1)where n > m > 1, p > q > 1, pm > qn; Ω is a bounded domain in RN with a smooth boundary ∂Ω; λ > 0 is a parameter, φ and ψ are nonnegative continuous functions on Ω .The existence and the uniqueness of local classical solutions to problem of semilinear parabolic systems are well known
We denote by Tλ∗ the maximal existence time of a classical solution (u, v) of problem (1.1), that is, Tλ∗ = sup T > 0, sup u(·, t) ∞ + v(·, t) ∞ < ∞, and we call Tλ∗ the life span of (u, v)
Friedman and Lacey [3] gave a result on the life span of solutions of (1.2) in the case of small diffusion

Summary

Introduction

We denote by Tλ∗ the maximal existence time of a classical solution (u, v) of problem (1.1), that is, Tλ∗ = sup T > 0, sup u(·, t) ∞ + v(·, t) ∞ < ∞ , and we call Tλ∗ the life span of (u, v). Friedman and Lacey [3] gave a result on the life span of solutions of (1.2) in the case of small diffusion. Gui and Wang [4], Lee and Ni [5] obtained the leading term of the expansion of the life span Tλ of the solution for (1.2) with the initial data λφ(x), and later, Mizoguchi and Yanagida [6] extended the result and determined the second term of the expansion of Tλ, the proved that φ attains the maximum at only one point as λ → ∞.

Objectives

Results

Conclusion