A problem with nonlocal conditions for partial differential equations with variable coefficients (Q2784884)

The authors consider the partial differential equation NEWLINE\[NEWLINE \prod_{j=1}^n\left(\frac{\partial}{\partial t}-a_j(t)L- b_j(t)\right)u(t,x)=f(t,x) NEWLINE\]NEWLINE under the conditions NEWLINE\[NEWLINE\frac{\partial^{q-1}u(t,x)}{\partial t^{q-1}}|_{t=0}- \mu\frac{\partial^{q-1}u(t,x)}{\partial t^{q-1}}|_{t=T}=\varphi_q(x), \qquad q=1,\ldots, n;NEWLINE\]NEWLINE NEWLINE\[NEWLINE L^su|_{x=0}=L^su|_{x=l}=0, \qquad s=0,1,\ldots,n-1, NEWLINE\]NEWLINE where \((t,x)\in D=\{(t,x)\in{\mathbb R}^2: t\in (0,T)\), \(x\in(0,l)\}\); \(a_j(t),b_j(t)\) are complex-valued functions, \(\mu\in\mathbb{C}\backslash \{0\}, \) \( L\equiv -\frac{d}{dx}(p(x)\frac{d}{dx})+q(x),\) \(p\in C^{2n-1}([0,l]),\) \(q\in C^{2n-2}([0,l]),\) \(p(x)\geq p_0>0, \) \( q(x)\geq 0.\) A solution to the problem is found as the Fourier series in eigenfunctions. Criteria of unique solvability of the problem are shown. The authors also consider the problem under the condition of periodicity \( a_i^{(\nu)}(0)=a_j^{(\nu)}(T),\) \(b_j^{(\nu)}(0)=b_j^{(\nu)}(T),\) \(\nu=0,1, \ldots, j-2\), \(j=2,3,\ldots,n.\) Conditions of existence of a solution to the problem are found. A theorem is proved about continuous dependence of solutions on the functions \(f(t,x)\) and \(\varphi_q(x),\) \(q=1,2,\ldots,n\).