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Uniform and optimal estimates for solutions to singularly perturbed parabolic equations. (English) Zbl 1130.35009
Let $$L$$ be the operator defined by $Lv=v_t+k(x,y,t)v_y-a(x,y,t)v_{xx}-b(x,y,t)v_x-c(x,y,t)v$ with $$a(x,y,t) \geq a_o>0$$. This paper deals with the boundary value problem for the equation $Lv=f(x,y,t) \tag{1}$ in $$\mathcal Q_T=\{(x,y,t)\in \mathbb R\times[0,1]\times [0,T]\}$$, with the y-periodic boundary condition $$v| _{y=0}=v| _{y=1}$$ and the initial data $$v(x,y,0)=\varphi(x,y).$$ The function $$f$$, and the coefficients of $$L$$ [resp. the initial data $$\varphi$$] are defined on $$\mathbb R\times \mathbb R\times [0,T]$$ [resp. $$\mathbb R^{2}],$$ smooth enough, and periodic in $$y$$ with the unit period. Equation (1) may be interpreted as an ultraparabolic equation with the time variables $$t_1=y,t_2=t$$, or as a parabolic equation, with space variables $$x,y,$$ degenerate with respect to $$y.$$ Under suitable assumptions on the coefficients of (1) and on $$f,\varphi$$, the authors prove the existence of a global in time solution of this problem, which belongs to the anisotropic Sobolev space $$W^{3,2,1}_2(\mathcal Q_{T})$$ as well as the anisotropic Hölder space $$C^{\lambda,\lambda,\frac{1}{12}}(\mathcal Q_T\cap\{x\in[-K,K]\})$$ for $$\lambda\in (0,1)$$ and $$K>0$$. Moreover , for every $$P\geq 0,$$ [resp. for every m=0,1,…] $$v$$ can be estimated as $$| v(x,y,t)| \leq C_Pe^{-P| x| }$$ , [resp. $$| v(x,y,t)| \leq \frac{C_m}{1+| x| ^m}]$$, in $$\mathcal Q_T$$ , according to the assumptions on the growth of the coefficients in $$L$$ . The proof is based on the obtention of $$\varepsilon$$ uniform estimates for the solution $$u^{\varepsilon}$$ to the regularized parabolic problem $Lu^{\varepsilon} -\varepsilon u_{yy}^{\varepsilon} =f(x,y,t)\quad \text{in } \mathcal Q_T$ $(u^{\varepsilon},u_y^{\varepsilon})| _{y=0}=(u^{\varepsilon},u_y^{\varepsilon})| _{y=1},\quad u^{\varepsilon}(x,y,0)=\varphi(x,y,)$ An application of these results to the solution $$\rho(\theta,\omega,t,\Omega)$$ to the Fokker-Plank type equation [cf. M. M. Lavrentiev jun. and the authors [Sib. Mat. Zh. 42, No. 4, 825–848 (2001; Zbl 0983.35075)] yields to the estimate $$| \rho(\theta,\omega,t,\Omega)| \leq Ce^{-M\omega^{2}}.$$ This paper completes the results obtained recently by the authors and M. M. Lavrentiev, in particular in [Asympt. Anal. 35, No. 1, 65–89 (2003; Zbl 1043.35023), Differ. Integral Equ., 17, No. 1–2, 99–118 (2004; Zbl 1164.35312)] and loc.cit.

##### MSC:
 35B25 Singular perturbations in context of PDEs 35K20 Initial-boundary value problems for second-order parabolic equations 35A05 General existence and uniqueness theorems (PDE) (MSC2000) 35K70 Ultraparabolic equations, pseudoparabolic equations, etc.
ASYMPT
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