zbMATH — the first resource for mathematics

On parabolic boundary value problems with a small parameter on the derivatives with respect to \(t\). (English. Russian original) Zbl 0639.35005
Math. USSR, Sb. 59, No. 2, 287-302 (1988); translation from Mat. Sb., Nov. Ser. 131(173), No. 3(11), 293-308 (1986).
Let \(\Omega\) be the bounded domain in \(\mathbb{R}^n \)with the boundary of the class \(C^{2m+\alpha}\), \(Q_T=\Omega \times [0,T]\), \(T=\text{const}>0\); \(\Gamma_T\) is the side surface of the cylinder \(Q_T\).
The boundary value problem is considered \[ \varepsilon D_tu^{\varepsilon} = \sum_{k<\vert\mu\vert\le n} H_{\mu}(x,t,D^k_xu^{\varepsilon})D_x^{\mu}u^{\varepsilon} + H_0(x, t, D^k_xu^{\varepsilon}), \tag{1} \]
\[ C_j(x,t,D_x)u^{\varepsilon}|_{\Gamma_T}=0,\quad j=1,\ldots, mN; \tag{2} \]
\[ u^{\varepsilon}|_{t=0}=u_0(x). \tag{3} \] Here \(H_0,u^{\varepsilon}\) are the \(N\)-dimensional vector-functions; \(H_{\mu}\) are the matrix \(N\times N\)-dimensional functions, \(\varepsilon >0\) is the small parameter, \(\mu\) is the multiindex; \(K\), \(m\) are the natural numbers such that \(K\le 2m-1\); \(C_j\) are the linear differential operators of the orders \(n_ j\), \(0\le n_ j\le 2m-1.\)
The conditions by which the problem (1)–(3) has a unique solution for small values of the parameter \(\varepsilon\) are formulated and estimates of the solution in Hölder norms are established. These estimates characterize the dependence of the speed of the convergence of the solution to the limit function as \(\varepsilon \to 0\) from the smoothness of the problem data.
Analogue results for the systems with the small multiplier at the part of derivatives with respect to \(t\) are received.

35B25 Singular perturbations in context of PDEs
35K52 Initial-boundary value problems for higher-order parabolic systems
35B45 A priori estimates in context of PDEs
Full Text: DOI