zbMATH — the first resource for mathematics

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.

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.
Full Text: DOI