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The delay effect on reaction-diffusion equations. (English) Zbl 1067.35139
The paper deals with the following nonlocal reaction-diffusion equation with delay $$\align u_t(t,x) &= \text{div}(v \nabla u) (t,x) + kf(u(t,x),u(t-\tau,x)),\\ u(t,x) &= 0, \quad t>0, \ x \in \partial \Omega,\\ u(t,x) &= \psi(t,x), \quad (t,x) \in [-\tau,0] \times \Omega, \ \psi \in C([-\tau,0] \times H_0^1(\Omega)), \endalign$$ where $v \in C^1({\overline \Omega},\Bbb R)$, $v(x) \geq m_0 >0$ for all $x \in {\overline \Omega}$, $\Omega \subset \Bbb R^m$ is a bounded domain with smooth boundary. The authors give conditions on the diffusion coefficient and the forcing term such that there exist a positive equilibrium of the above problem as well as periodic solutions.

35R10Partial functional-differential equations
35B32Bifurcation (PDE)
35K57Reaction-diffusion equations
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