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The Riemann-Hilbert boundary value problem for semilinear pseudoparabolic equations. (English) Zbl 0814.35090

The author considers a Riemann-Hilbert type initial boundary value problem for semilinear pseudoparabolic equations of the form \[ \begin{split} W_{\overline zt} + q_ 1 W_{zt} + a_ 2 \overline W_{zt} + a_ 1 W_ t + a_ 2 \\ \overline W_ t + b_ 1 W_{\overline z} + b_ 2 \overline W_{\overline z} + b_ 3 \overline W_ z + b_ 4 \overline W_ z + H(t,t,W) = 0, \end{split} \] in \(D \times I\) where \(D\) is a simply connected domain with continuously differentiable boundary \(\Gamma\) in the complex plane \(\mathbb{C}\) and \(I\) is an interval on the real line \(\mathbb{R}\).
Under some conditions concerning the functions \(q_ 1, q_ 2\), \(a_ 1, a_ 2\), \(b_ i\) \((i=1,4)\), \(H\), and the functions involved in the initial data, it is proved that the considered Riemann-Hilbert problem has at least one solution \(W\) defined on \(D \times [0,T^*]\), such that \(W \in C^{0,1} (I^*, W_ 0^{1,p} (D))\), where \(I^* = [0,T^*]\) is a subset of \(I\), \(2 < p_ 0 \leq p\), and \(p\) is a number fixed in one of the imposed condition.

MSC:

35Q15 Riemann-Hilbert problems in context of PDEs
35K70 Ultraparabolic equations, pseudoparabolic equations, etc.
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