Semilinear pseudo-parabolic equations on manifolds with conical singularities

Yitian Wang; Xiaoping Liu; Yuxuan Chen; Yitian Wang; Xiaoping Liu; Yuxuan Chen

doi:10.3934/era.2021057

Electronic Research Archive

2021, Volume 29, Issue 6: 3687-3720. doi: 10.3934/era.2021057

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Semilinear pseudo-parabolic equations on manifolds with conical singularities

Yitian Wang ^{1
,},
Xiaoping Liu ^{1
,},
Yuxuan Chen ^{1,2
,
,}

1.
College of Mathematical Sciences, Harbin Engineering University, 150001, China
2.
College of Intelligent Systems Science and Engineering, Harbin Engineering University, 150001, China

Received: 01 March 2021 Revised: 01 May 2021 Published: 13 August 2021
35K20, 35K55, 35A01, 35D30

This paper studies the well-posedness of the semilinear pseudo-parabolic equations on manifolds with conical degeneration. By employing the Galerkin method and performing energy estimates, we first establish the local-in-time well-posedness of the solution. Moreover, to reveal the relationship between the initial datum and the global-in-time well-posedness of the solution we divide the initial datum into three classes by the potential well depth, i.e., the sub-critical initial energy level, the critical initial energy level and the sup-critical initial energy level (included in the arbitrary high initial energy case), and finally we give an affirmative answer to the question whether the solution exists globally or not. For the sub-critical and critical initial energy, thanks to the potential well theory, we not only obtain the invariant manifolds, global existence and asymptotic behavior of solutions, but also prove the finite time blow up of solutions and estimate the lower bound the of blowup time. For the sup-critical case, we show the assumptions for initial datum which cause the finite time blowup of the solution, realized by introducing a new auxiliary function. Additionally, we also provide some results concerning the estimates of the upper bound of the blowup time in the sup-critical initial energy.
- Conical Laplacian,
- semilinear pseudo-parabolic,
- global existence,
- finite time blow up,
- blowup time,
- asymptotic behavior
Citation: Yitian Wang, Xiaoping Liu, Yuxuan Chen. Semilinear pseudo-parabolic equations on manifolds with conical singularities[J]. Electronic Research Archive, 2021, 29(6): 3687-3720. doi: 10.3934/era.2021057

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Abstract

This paper studies the well-posedness of the semilinear pseudo-parabolic equations on manifolds with conical degeneration. By employing the Galerkin method and performing energy estimates, we first establish the local-in-time well-posedness of the solution. Moreover, to reveal the relationship between the initial datum and the global-in-time well-posedness of the solution we divide the initial datum into three classes by the potential well depth, i.e., the sub-critical initial energy level, the critical initial energy level and the sup-critical initial energy level (included in the arbitrary high initial energy case), and finally we give an affirmative answer to the question whether the solution exists globally or not. For the sub-critical and critical initial energy, thanks to the potential well theory, we not only obtain the invariant manifolds, global existence and asymptotic behavior of solutions, but also prove the finite time blow up of solutions and estimate the lower bound the of blowup time. For the sup-critical case, we show the assumptions for initial datum which cause the finite time blowup of the solution, realized by introducing a new auxiliary function. Additionally, we also provide some results concerning the estimates of the upper bound of the blowup time in the sup-critical initial energy.

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