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Global existence of solution of Cauchy problem for nonlinear pseudo-parabolic equation. (English) Zbl 1156.35414

Summary: We prove that the Cauchy problem for the nonlinear pseudo-parabolic equation \[ v_t - \alpha v_{xxt} - \beta v_{xx}+\gamma v_x+f(v)_x=\varphi (v_x)_x+g(v) - \alpha g(v)_{xx} \]
admits a unique global generalized solution in \(C^{1}([0,\infty );H^s(\mathbb R))\), a unique global classical solution and asymptotic behavior of the solution. We also prove that the Cauchy problem for the equation
\[ v_t - \alpha v_{xxt} - \beta v_{xx}=g(v) - \alpha g(v)_{xx} \]
has a unique global generalized solution in \(C^{1}([0,\infty );W^{m,p}(\mathbb R)\cap L^{\infty }(\mathbb R))\), a unique global classical solution and asymptotic behavior of the solution.

MSC:

35K70 Ultraparabolic equations, pseudoparabolic equations, etc.
35K25 Higher-order parabolic equations
35K55 Nonlinear parabolic equations
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