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The local strong and weak solutions for a generalized pseudoparabolic equation. (English) Zbl 1246.35113

Summary: The Cauchy problem for a nonlinear generalized pseudoparabolic equation is investigated. The well-posedness of local strong solutions for the problem is established in the Sobolev space \(C([0, T); H^s(\mathbb R)) \bigcap C^1([0, T); H^{s-1}(\mathbb R))\) with \(s > 3/2\), while the existence of local weak solutions is proved in the space \(H^s(\mathbb R)\) with \(1 \leq s \leq 3/2\). Further, under certain assumptions of the nonlinear terms in the equation, it is shown that there exists a unique global strong solution to the problem in the space \(C([0, \infty); H^s(\mathbb R)) \bigcap C^1([0, \infty); H^{s-1}(\mathbb R))\) with \(s \geq 2\).

MSC:

35K70 Ultraparabolic equations, pseudoparabolic equations, etc.
35K15 Initial value problems for second-order parabolic equations
35K55 Nonlinear parabolic equations
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