Abstract
In this paper, we consider a system of two heat equations with nonlinear boundary flux which obey different laws, one is exponential nonlinearity and another is power nonlinearity. Under certain hypotheses on the initial data, we get the sufficient and necessary conditions, on which there exist initial data such that non-simultaneous blow-up occurs. Moreover, we get some conditions on which simultaneous blow-up must occur. Furthermore, we also get a result on the coexistence of both simultaneous and non-simultaneous blow-ups.
Highlights
1 Introduction and main results In this paper, we study the following system of two heat equations coupled by nonlinear boundary conditions
3 Non-simultaneous blow-up we prove Theorem . with four lemmas
Author details 1Department of Mathematics, Jiangxi Vocational College of Finance and Economics, Jiujiang, Jiangxi 332000, P.R. China
Summary
Let (u, v) be a solution of the following system:. ) blow up with large initial data if α > , or β > , or pq > β(α – ). By the comparison principle, (u, v) is a sub-solution of J(x, ) = u – εuα epv ≥ , x ∈ BR, K (x, ) = v – εuq eβv ≥ , x ∈ BR. The following lemma proves the sufficient and necessary condition on the existence of u blowing up alone. There exist suitable initial data such that u blows up while v remains bounded if and only if α > q +. Assume (u , v ) is a pair of initial data such that the solution of Let the minimum of u (≥ u ) be large such that T is small and satisfies
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