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Existence and asymptotic stability of periodic solutions with an interior layer of reaction-advection-diffusion equations. (English) Zbl 1325.35099

The paper deals with the singularly perturbed periodic boundary value problem
\[ \begin{cases} N_\varepsilon (u):=\varepsilon\left(\dfrac{\partial^2 u}{\partial x^2}-\dfrac{\partial u}{\partial t}\right)-A(u,x,t)\dfrac{\partial u}{\partial x}-B(u,x,t)=0,\\ \quad\quad \text{for }(x,t)\in\mathcal{D}:=\{(x,t)\in\mathbb R^2: -1<x<1, t\in\mathbb R\},\\ u(-1,t,\varepsilon)=u^{(-)}(t),\, u(1,t,\varepsilon)=u^{(+)}(t),\quad \text{for }t\in\mathbb R,\\ u(x,t,\varepsilon)=u(x,t+T,\varepsilon),\quad \text{for }t\in\mathbb R\quad -1\leq x\leq 1,\end{cases} \]
for \(\varepsilon\in I_{\varepsilon_0}:=\{0<\varepsilon\leq \varepsilon_0\}\), \(0<\varepsilon_0\ll 1\), and for functions \(A\), \(B\), \(u^{(-)}\) and \(u^{(+)}\) sufficiently smooth and \(T\)-periodic in \(t\). By constructing the asymptotic lower and upper solutions, the authors prove the existence of a periodic solution of the above with an interior layer with respect to \(x\), and estimate the accuracy of its asymptotics. The asymptotic stability of this solution is also established.

MSC:

35K57 Reaction-diffusion equations
35K20 Initial-boundary value problems for second-order parabolic equations
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