Asymptotic expansions on the small parameter for solutions of linear symmetric systems (Q2748969)

Consider the following Cauchy problem NEWLINE\[NEWLINE (P_0+\varepsilon P_1)U =F(x,t),\quad x\in \mathbb{R}^d,\;t>0,\qquad U(\varepsilon,x,0)=U(x),\quad x\in \mathbb{R}, \tag{1} NEWLINE\]NEWLINE where \(P_i =(A_i\partial _t+B_i(x)\partial _x)+G_i(x)\), \(i=0,1\), \(B_i\), \(G_i\) are real \(n\times n\) matrices, \(d\geq 1\), \(\varepsilon >0\) is a small parameter, \(U,F:\mathbb{R}\times [0,\infty)\to \mathbb{R}^n\), NEWLINE\[NEWLINE A_0=\left(\begin{matrix} I_m & 0\\ 0 & 0 \end{matrix} \right),\quad A_1=\left(\begin{matrix} 0 & 0\\ 0 & I_{n-m} \end{matrix} \right),\quad \quad 0\leq m\leq n, NEWLINE\]NEWLINE and \(I_k\) is a unity matrix. The author studies the behavior of solution \(U(\varepsilon,x,t)\) to the problem \((1)\) as \(\varepsilon\to 0\). For \(s\in N\) denote by \(H^s\) the usual Sobolev spaces with the norm \(\|\cdot\|_s\), \(H^s_n =(H^s)^n\) is the Hilbert space equipped with the scalar product \((f_1,f_2)_{s,n}=\sum _{j=1}^n (f_{1j},f_{2j})_s\), \(f_i =(f_{i1},\dots ,f_{in})\), \(i=1,2\) and with the norm \(\|\cdot \|_{s,n}\) generated by this scalar product. For \(k\in N^*\) and \(1\leq p \leq \infty\) and Banach space \(X\) set \(W^{k,p}(a,b;X)=\{u\in{\mathcal D}'((a,b);X)\); \(u^{(j)}\in L^p(a,b;X)\), \(j=0,1,\dots, k\}\), where \(u^{(j)}\) is the distributional derivative of order \(j\). Denote by \(B=B_0+\varepsilon B_1\), \(G=G_0+\varepsilon G_1\). The special forms of matrices \(A_0\) and \(A_1\) involve the natural representations of matrices \(B_i,G_i\) by blocks NEWLINE\[NEWLINE B_j = \left(\begin{matrix} B_{j1} & B_{j2}\\ B^*_{j2} & B_{j3} \end{matrix} \right),\qquad G_j = \left(\begin{matrix} G_{j1} & G_{j2}\\ G^*_{j2} & G_{j3} \end{matrix} \right),\qquad j=0,1, NEWLINE\]NEWLINE where \(B_{j1}, G_{j1} \in M^m(\mathbb{R})\), \(B_{j2},G_{j2} \in M^{m\times (n-m)}(\mathbb{R})\), \(B_{j3},G_{j3} \in M^{n-m}(\mathbb{R})\), and ``*'' means the transposition. Suppose that \(B\) and \(G\) satisfy the following conditions: \(B_i(x), G_i(x)\), \(i=0,1,\) are real symmetric matrices for \(x\in \mathbb{R}\); \((G(x)\xi-1/2\partial_xB(x)\xi,\xi)_{\mathbb{R}^n}\geq q_0|\eta |^2\); \((G_{03}(x)\eta-1/2\partial_xB_{03}(x)\eta,\eta)_{\mathbb{R}^n}\geq q_0|\eta |^2\), \(q_0>0\); for all \(\xi =(\xi',\eta)\in \mathbb{R}^n\) and \( \eta \in \mathbb{R}^{n-m}\); \(|\det B_{03}(x)|\geq b_0>0\), for all \(x\in \mathbb{R}\). Using the method of Lyusternik-Vichik, for the solution \(U(\varepsilon,x,t)\) to the problem (1) is postulated the following asymptotic expansion NEWLINE\[NEWLINE U(\varepsilon,x,t) = \sum_{k=0}^N \varepsilon^k(V_k(x,t)+Z_k(x,\tau))+ R_{N}(\varepsilon,x,t),\quad \tau=\frac {t}{\varepsilon},\qquad \tag{2} NEWLINE\]NEWLINE where \(Z(x,\tau)=Z_0(x,\tau)+\cdots + \varepsilon^NZ_N(x,\tau)\) is the boundary layer function. It describes the singular behavior of solution \(U(\varepsilon,x,t)\) relative to \(\varepsilon\) within a neighborhood of the set \(\{(x,0),x\in \mathbb{R}\}\). The function \(V(x,t)=V_0(x,t)+ \cdots + \varepsilon^N V_N(x,t)\) is the regular part of expansion \((2)\). NEWLINENEWLINENEWLINEThe main result of the paper is following. Suppose \(0\leq l<N+1\). If \(U_0 \in H^{s+2l+3(N+1)}_n\), \(F\in W^{l+1,1}(0,T;H^{s+2l+3(N+1)}_n)\), then there exists a unique strong solution \(U \in W^{l,\infty}(0,T;H^s_n)\) of the problem \((1)\). For this solution is valid expansion (2) , where \(V_k\) and \(Z_k\) satisfy the estimates NEWLINE\[NEWLINE \|V_k \|_{W^{l,\infty}(0,T;H^s_n)}\leq C(T)(\|U_0\|_{s+2l+3k+1,n}+ \|F(\cdot,0)\|_{s+2l+3k-2,n}+\|F \|_{W^{l,1}(0,T;H^{s+3k+2}_n)}), NEWLINE\]NEWLINE NEWLINE\[NEWLINE \|\partial _{\tau} ^lZ_k(\cdot,\tau) \|_{s,n}\leq C e^{-q_0\tau} (1+\tau^k)(\|U_0\|_{s+l+k+1,n}+ \|F(\cdot,0)\|_{s+l+k,n}). NEWLINE\]NEWLINE For the remainder term \(R_N= \text{col}(R_{N1},R_{N2})\) is true the estimate NEWLINE\[NEWLINE \|R_{N1}\|^2_{W^{l,\infty}(0,T;H^s_m)} +\varepsilon^{\frac 12} \|R_{N2}\|^2_{W^{l,\infty}(0,T;H^s_{n-m})}\leq C(T)\varepsilon^{N+1-l}.NEWLINE\]