On solvability of boundary value problems for non-Cauchy-Kovalevskaya-type systems (Q2760737)

The author studies the solvability of boundary value problems for higher-order systems with constant coefficients NEWLINE\[NEWLINE \begin{aligned} KD_t u^+ +K_1 u^+ +K_2 \int\limits_0^t u^+(x,\tau)\,d\tau &+ L(D_x)u^-(x,t)=0, \quad t>0,\;x\in \mathbb R_n^+=\{x\in \mathbb R_n:\;x_n>0\}, \tag{1} \\ M(D_x)u^+&=0, \tag{2} \\ Bu^+(x,t)| _{x_n=0}&=0\;(t>0),\;\;u^+| _{t=0}=u_0(x) \;(x\in \mathbb R_n^+), \tag{3} \end{aligned} NEWLINE\]NEWLINE where \(M\) and \(L\) are matrix differential operators with constant coefficients, \(K,K_1,K_2\) are constant matrices, and \(B\) is a matrix boundary operator. The vector-valued functions \(u^{+}, u^-\) are of length \(m\) and \(\nu-m\), respectively. It is assumed that the above system possesses certain homogeneity properties and the Lopatinskiĭ\ condition holds; i.e., \(B\) is a \(\mu\times m\)-matrix and the boundary value problem with parameters NEWLINE\[NEWLINE \begin{gathered} (\tau K + K_1 +\tau^{-1}K_2)v^+ + L(i\xi',D_{x_n})v^- =0, \;\;M(i\xi',D_{x_n})v^+=0\;\;(x_n>0), \\ Bv^+| _{x_n=0}=\psi,\;\;\sup\limits_{x_n>0}| v(\tau,\xi',x_n)| <\infty \end{gathered} NEWLINE\]NEWLINE is uniquely solvable for \(\text{Re}\,\tau\geq \gamma_0,\) \(\xi'\in R_{n-1}\setminus \{0\}\), and an arbitrary vector \(\psi=(\psi_1,\psi_2,\dots,\psi_\mu)\). A solution \(u=(u^+,u^-)\) to the initial boundary value problem (1)--(3) is sought in the Sobolev class \(e^{\gamma t}u=e^{\gamma t}(u^+,u^-)\in W_{p}^{l,r}(R_{n+1}^{++})\) (\(\gamma> \gamma_0\)), where \(R_{n+1}^{++}=\{(x,t):\;x\in R_n, t>0, x_n>0\}\), \(l\geq 0\) is an integer and \(r=(r_1,r_2,\dots,r_n)\) is a vector of nonnegative integers. Thus, the symbol \(W_{p}^{l,r}(R_{n+1}^{++})\) denotes the conventional anisotropic Sobolev space. The article contains several solvability and uniqueness theorems for boundary value problems (1)--(3).