An ill-posed nonlocal two-point problem for systems of partial differential equations. (Russian, English) Zbl 1081.35131

Il’kiv, V. S.; Ptashnik, B. I.

Sib. Mat. Zh. 46, No. 1, 119-129 (2005); translation in Sib. Math. J. 46, No. 1, 94-102 (2005).

Let \(Q = (0,T)\times \Omega\), where \(\Omega\) is a \(p\)-dimensional torus. The following nonlocal problem is studied in \(Q\): \[ \begin{gathered} \frac{\partial^n u}{\partial t^n} = \sum_{j=1}^nL_j(t,D)\frac{\partial^{n-j}u}{\partial t^{n-j}} +f,\\ lu\equiv \sum_{j=1}^n\left(B_{0j}(D)\left.\frac{\partial^{n-j}u}{\partial t^{n-j}}\right| _{t=0} + B_{Tj}(D)\left.\frac{\partial^{n-j}u}{\partial t^{n-j}}\right| _{t=T}\right) = \varphi, \end{gathered} \] where \(L_j(t,D) \equiv \sum_{| s| \leq j}a_{js}(t)D^s\), \(B_{0j}(D) \equiv \sum_{| s| \leq j}B_{0js}D^s\), \(B_{Tj}(D) \equiv \sum_{| s| \leq j}B_{Tjs}D^s\), \(D^s \equiv D_1^{s_1}\dots D_p^{s_p}\), \(D_j \equiv -i\partial/ \partial x_j\), \(j=1,\dots,p\), \(| s| = s_1 +\dots + s_p\), \(B_{0js}\) and \(B_{Tjs}\) are (\(nm\times m\))-dimensional complex matrix-valued coefficients, and \(f\equiv f(t,x)\), \(\varphi \equiv \varphi(x)\) are given vector-valued functions. The \(m\)-dimensional complex matrix-valued coefficients \(a_{js}(t)\) are assumed to be continuous on \([0,T]\).
The authors prove the existence of a so-called pseudosolution to the above problem and establish conditions which guarantee the uniqueness of the solution obtained.

Reviewer: V. Grebenev (Novosibirsk)

MSC:

35R25	Ill-posed problems for PDEs
35A05	General existence and uniqueness theorems (PDE) (MSC2000)
46E35	Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
49J35	Existence of solutions for minimax problems

Keywords:

pseudosolution; convex minimization problem; existence; uniqueness