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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 \(\varepsilon \to 0\).

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
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