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Some non-local problems for the parabolic-hyperbolic type equation with complex spectral parameter. (English) Zbl 1158.35390

The equation $Lu=su$ is under consideration, where

$Lu=\left\{\begin{array}{cc}{u}_{xx}-{u}_{y},\phantom{\rule{4pt}{0ex}}\hfill & y>0,\hfill \\ {u}_{xx}-{u}_{yy},\phantom{\rule{4pt}{0ex}}\hfill & y<0,\hfill \end{array}\right\$

$s=\lambda$ at $y>0$ and $s=-\mu$ at $y<0$, $\lambda$, $\mu$ are given complex numbers. Let ${\Omega }$ be a simple connected domain, located in the plane of independent variables $x$, $y$, at $y>0$ sectionally bounded $A{A}_{0}$, $B{B}_{0}$, ${A}_{0}{B}_{0}\left(A\left(0,0\right),B\left(1,0\right),{A}_{0}\left(0,1\right),{B}_{0}\left(l,1\right)\right)$ and at $y<0$ bounded with characteristics $AC:x+y=0$, $BC:x-y=1$ of the above equation. The following notations are introduced:

${{\Omega }}_{1}:={\Omega }\cap \left\{y>0\right\}$, ${{\Omega }}_{2}:={\Omega }\cap \left\{y<0\right\}$, $AB:=\left\{\left(x,y\right):y=0,\phantom{\rule{0.166667em}{0ex}}0, ${{\Theta }}_{0}$ and ${\theta }_{1}$ are points of intersection of characteristics, which are outgoing from the point $\left(x,0\right)\in AB$ with characteristics $AC$, $BC$ respectively, i.e.,

${\theta }_{0}=\left(\frac{x}{2},-\frac{x}{2}\right),\phantom{\rule{1.em}{0ex}}{\theta }_{1}=\left(\frac{x+1}{2},-\frac{x-1}{2}\right)·$

The following non-local problems are defined:

Problem ${S}_{1}$: Find a regular solution of the above equation in ${\Omega }$ satisfying the conditions

${u\left(x,y\right)|}_{A{A}_{0}}={\varphi }_{1}{\left(y\right),\phantom{\rule{1.em}{0ex}}u\left(x,y\right)|}_{B{B}_{0}}={\varphi }_{2}\left(y\right),\phantom{\rule{1.em}{0ex}}0\le y\le 1,$
${a}_{1}\left(x\right){A}_{0x}^{0,\sqrt{\mu }}\left[u\left({0}_{0}\right)\right]+{b}_{1}\left(x\right){A}_{1x}^{1,\sqrt{\mu }}\left[u\left({0}_{1}\right)\right]+{c}_{1}\left(x\right)u\left(x,0\right)={d}_{1}\left(x\right),\phantom{\rule{1.em}{0ex}}x\in \overline{AB}·$

Problem ${S}_{2}$: Find regular solution of the above equation in ${\Omega }$ satisfying the condition the first conditionn in Problem ${S}_{1}$ and

${a}_{2}\left(x\right){A}_{0x}^{1,\sqrt{\mu }}\left[\frac{d}{dx}u\left({\theta }_{0}\right)\right]+{b}_{2}\left(x\right){A}_{1x}^{1,\sqrt{\mu }}\left[\frac{d}{dx}u\left({\theta }_{1}\right)\right]+{c}_{2}\left(x\right){u}_{y}\left(x,0\right)={d}_{2}\left(x\right),\phantom{\rule{1.em}{0ex}}x\in AB·$

Here ${a}_{i}\left(x\right)$, ${b}_{i}\left(x\right)$, ${c}_{i}\left(x\right)$ $\left(i=1,2\right)$ are given real-valued functions, moreover ${a}_{i}^{2}\left(x\right)+{b}_{i}^{2}\left(x\right)\ne 0$, for all $x\in \left[0,1\right]$ and ${\varphi }_{1}\left(y\right)$, ${\varphi }_{2}\left(y\right)$, ${d}_{i}\left(x\right)$ are, generally speaking, complex-valued functions. Here $\sqrt{\mu }$ is the square root of the complex number $\mu$ with $|arg\left(\mu \right)|\le \pi$.

In the present-paper the unique solvability of the both non-local problems ${S}_{1}$ and ${S}_{2}$ for the considered mixed parabolic-hyperbolic type equation with complex spectral parameter is proved. Further sectors for values of the spectral parameter where these problems have unique solutions are shown. Uniqueness of the solution is proved by the method of energy integral and existence is proved by the method of integral equations. In particular cases, eigenvalues and corresponding eigenfunctions of the studied problems are found.

MSC:
 35M10 PDE of mixed type 35P05 General topics in linear spectral theory of PDE