Abstract
In this paper, we study the blow-up property of positive mild solutions to the Cauchy problem of a system of fractional reaction-diffusion equations. For the fundamental solution \(P(t,x)\) of the fractional heat operator \(\partial_{t}+(-\triangle)^{\frac{\beta}{2}}\) defined on the whole space \(R^{N}\), due to the properties of \(P(t,x)\) established by H Yosida and some estimates of \(P(t,x)\) developed by L Caffarelli and A Figalli, we first use an iteration method to establish the estimates of lower bounds of positive mild solutions; then we obtain the unboundedness of solutions for large time. Finally we give a sufficient condition that the positive mild solution to a fractional reaction-diffusion system blows up in finite time.
Highlights
In this paper, we consider the blow-up solutions to the Cauchy problem of a system of fractional reaction-diffusion equations, ⎧ ⎪⎨ut + (– ) β u = vp, x ∈ RN, t >, ⎪⎩vt + u(, (– x)) v= = u (x) uq ≥, x ∈ RN, t >, v(, x) = v (x) ≥, x ∈ RN
In Section, we prove the blow-up property of the positive mild solution to the problem ( . )
4 Discussion In this paper, we investigated the blow-up property of the positive mild solutions to a system of fractional reaction-diffusion equations
Summary
We consider the blow-up solutions to the Cauchy problem of a system of fractional reaction-diffusion equations,. Wang and Wang [ ] considered the nonlocal equation ut – u = f u(t, x) dx, x ∈ , t > , and Chadam et al [ ] considered the equation with a localized source term ut – u = f u(t, x ) , x , x ∈ , t > , with homogeneous Neumann boundary values and nonnegative initial data Under appropriate conditions, they proved that the solution blows up in finite time and the blow-up is set in , respectively. We give a sufficient condition for the positive mild solution to a fractional reaction-diffusion system to blow up in finite time. In Section , we give a discussion of the study of systems of fractional reaction-diffusion equations
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