## Dirichlet problem for the diffusive Nicholson’s blowflies equation.(English)Zbl 0923.35195

The equation considered in this paper is the diffusive Nicholson’s blowflies equation $\partial u(t,x)/\partial t= d\Delta u(t,x)- 2\tau u(t,x)+ \beta\tau u(t- 1,x) e^{-u(t-1,x)},\tag{1}$ where $$(t,x)\in D\equiv (0,\infty)\times\Omega$$, $$\Omega\subseteq \mathbb{R}^n$$ $$(n\geq 1)$$ being a bounded domain with smooth boundary $$\partial\Omega$$; $$\beta$$, $$\tau$$, $$d$$ are positive constants.
The aim of this paper is to study the asymptotic behavior of solutions of the equation (1) under Dirichlet condition $u\equiv 0\quad\text{on }\Gamma= (0,\infty)\times \partial\Omega\tag{2}$ and initial condition $u(\theta,x)= u_0(\theta, x)\quad\text{in }D_1\equiv [-1,0]\times\overline\Omega.\tag{3}$ A new approach is developed to study the global attractivity of the positive solution of the problem $d\Delta\phi(x)- \tau\phi(x)+ \beta\tau\phi(x) \exp(-\phi(x))= 0\quad\text{for }x\in\Omega,\quad \phi(x)= 0\quad\text{for }x\in\partial\Omega.\tag{4}$ A main result is the following: If $$e\leq \beta\leq e^2$$ and $$u(t,x)$$ is a nontrivial and nonnegative solution of (1)–(2) and $$\phi(x)$$ is the positive solution of (4), then $\lim_{t\to\infty} \| u(t,.)- (.)\|_{C(\overline\Omega)}= 0\quad\text{and} \quad \lim_{t\to\infty} \| u(t,.)- \phi(.)\|_{L^2(\Omega)}= 0.$

### MSC:

 35R10 Partial functional-differential equations 35B40 Asymptotic behavior of solutions to PDEs
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### References:

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