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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

${v}_{t}-\alpha {v}_{xxt}-\beta {v}_{xx}+\gamma {v}_{x}+f{\left(v\right)}_{x}=\phi {\left({v}_{x}\right)}_{x}+g\left(v\right)-\alpha g{\left(v\right)}_{xx}$

admits a unique global generalized solution in ${C}^{1}\left(\left[0,\infty \right);{H}^{s}\left(ℝ\right)\right)$, a unique global classical solution and asymptotic behavior of the solution. We also prove that the Cauchy problem for the equation

${v}_{t}-\alpha {v}_{xxt}-\beta {v}_{xx}=g\left(v\right)-\alpha g{\left(v\right)}_{xx}$

has a unique global generalized solution in ${C}^{1}\left(\left[0,\infty \right);{W}^{m,p}\left(ℝ\right)\cap {L}^{\infty }\left(ℝ\right)\right)$, a unique global classical solution and asymptotic behavior of the solution.

##### MSC:
 35K70 Ultraparabolic equations, pseudoparabolic equations, etc. 35K25 Higher order parabolic equations, general 35K55 Nonlinear parabolic equations
##### References:
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