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Boundary layer for chaffee-infante type equation. (English) Zbl 0911.35012
The paper deals with the following problem $\frac {\partial {u^\varepsilon }}{\partial t}-\varepsilon \Delta u^\varepsilon +\left (u^\varepsilon \right)^3 - u^\varepsilon =f$ for $$(x,y)\in \Omega =(0,2\pi)\times (0,1)$$, $$t\in I=(0,T)$$ with initial condition $$u^\varepsilon =u_0$$ for $$t=0$$ and boundary condition $$u^\varepsilon (x,0)=u^\varepsilon (x,1)=0$$. In $$x$$ condition of $$2\pi$$ periodicity is assumed for solutions and all data. The corresponding “inviscid” problem is $\frac {\partial {u^0}}{\partial t} + \left (u^0\right)^3 - u^0=f$ completed with the same initial and boundary conditions.
As $$\varepsilon \to 0$$ solutions $$u^\varepsilon$$ converge to $$u^0$$ inside domain $$\Omega$$ except for boundary layers adjacent to lines $$y=0$$ and $$y=1$$. In the paper correctors $$M,N$$ are “explicitly” constructed such that estimates $\left \| u^\varepsilon -u^0\right \| \leq \varepsilon ^\alpha$ in norm of $$L^\infty (I\times \Omega)$$ with $$\alpha =\frac 12$$ in norm of $$L^\infty (I,L^2(\Omega))$$ with $$\alpha =\frac 34$$ and in norm of $$L^\infty (I,H^1(\Omega))$$ with $$\alpha = \frac 14$$ hold as $$\varepsilon \to 0$$.
Reviewer: J.Franců (Brno)

##### MSC:
 35B25 Singular perturbations in context of PDEs 76D10 Boundary-layer theory, separation and reattachment, higher-order effects 35B40 Asymptotic behavior of solutions to PDEs 35C20 Asymptotic expansions of solutions to PDEs
##### Keywords:
corrector; nonlinear diffusion equation
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