Asymptotic analysis of an advection-diffusion equation involving interacting boundary and internal layers

As $\epsilon$ goes to zero, the unique solution of the scalar advection-diffusion equation $y^{\varepsilon}t-\varepsilon y^{\varepsilon}{xx} + M y^{\varepsilon}x=0$,$(x,t)\in (0,1)\times (0,T)$submittedtoDirichletboundaryconditionsexhibitsaboundarylayerofsize$$\mathcal{O}$(\varepsilon)$andaninternallayerofsize$$\mathcal{O}$(\sqrt{\varepsilon})$\mathrm{.}Ifthetime$T$ is large enough, these thin layers where the solution $y^{\epsilon }$ displays rapid variations intersect and interact each other. Using the method of matched asymptotic expansions, we show how we can construct an explicit approximation ${\tilde{P}}^{\epsilon }$ of the solution $y^{\epsilon }$ satisfying $\Vert y^{\varepsilon}-\widetilde{P}^\varepsilon\Vert{L^\infty(0,T; L^2(0,1))}=$\mathcal{O}$(\varepsilon^{3/2})$and$\Vert y^{\varepsilon}-\widetilde{P}^\varepsilon\Vert_{L^2(0,T; H^1(0,1))}=$\mathcal{O}$(\varepsilon)$,forall$\varepsilon$ small enough.