Blow-up for a damped p-Laplacian type wave equation with logarithmic nonlinearity

In this paper, we investigate the existence, uniqueness, exponential decay, and blow-up behavior of the viscoelastic beam equation involving the p-Laplacian operator, strong damping, and a logarithmic source term, given bywhere Ω is a bounded domain of R n and g > 0 is a memory kernel. Using the Faedo-Galerkin approximation, we establish the existence and uniqueness result for the global solutions, taking into account that the initial data must belong to an appropriate stability set created from the Nehari manifold. The study of the exponential decay of our problem is based on Nakao's method. Finally, the blow-up behavior on the instability set is proved.

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“…More recently, Yang and Han 27 proved the blow‐up in finite time of the solutions of the IBVP associated with the equation

{u}_{tt}-{\Delta}_pu-\Delta {u}_t&#x0002B;{u}_t&#x0003D;{\left&#x0007C;u\right&#x0007C;}&#x0005E;{p-2}u\ln \left&#x0007C;u\right&#x0007C;,

with initial data at different initial energy levels.…”

Section: Introductionmentioning

confidence: 99%

“…More recently, Yang and Han 27 proved the blow-up in finite time of the solutions of the IBVP associated with the equation…”

Section: Introductionmentioning

confidence: 99%

Blow‐up results for a viscoelastic beam equation of p‐Laplacian type with strong damping and logarithmic source

Pereira

Araújo

Raposo

et al. 2023

Math Methods in App Sciences

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A critical Kirchhoff equation with a logarithmic type perturbation in dimension 4

Han

2024

Math Methods in App Sciences

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In this paper, a critical Kirchhoff type elliptic equation involving a logarithmic type perturbation is considered in a bounded domain in . Three main features of the problem bring essential difficulties when proving the existence of weak solutions. The first one is the nonlocal term which makes the structure of the corresponding energy functional more complicated, the second one is that the problem is critical in the sense that the energy functional lacks compactness, and the third one is the appearance of the logarithmic term which satisfies neither the standard monotonicity condition nor the Ambrosetti–Rabinowitz condition. Moreover, the boundedness of the sequence is hard to obtain for Kirchhoff problems in dimension 4. By combining a result by Jeanjean (Proc. Roy. Soc. Edinburgh Sect. A, 1999, 129: 787–809) and a recent estimate by Deng et al. (Adv. Nonlinear Stud., 2023, 23: No. 20220049) with the mountain pass lemma and Brézis‐Lieb's lemma, it is proved that either the norm of the sequence of approximation solution goes to infinity or the problem admits a nontrivial weak solution, under appropriate assumptions on the parameters.

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Critical exponent for global solutions in a fourth‐order hyperbolic equation of p‐Kirchhoff type

Liu,

Dou

2024

Math Methods in App Sciences

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We study a fourth‐order hyperbolic equation involving Kirchhoff type ‐Laplacian and superlinear source, subject to zero Navier boundary condition, where is an open bounded domain in with ; denotes the maximal existence time; and and are constants. For , using auxiliary function method and Sobolev inequality, we prove that there are only global solutions. For , we obtain the optimal classification of initial energy and Nehari energy, which guarantees the existence of blow‐up solutions and global solutions. In the critical case , we find out that the coefficients of the Kirchhoff term and the superlinear source play important role in separating out the property of weak solutions.

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Blow-up for a damped p-Laplacian type wave equation with logarithmic nonlinearity

Cited by 20 publications

References 16 publications

Blow‐up results for a viscoelastic beam equation of p‐Laplacian type with strong damping and logarithmic source

Blow‐up results for a viscoelastic beam equation of p‐Laplacian type with strong damping and logarithmic source

A critical Kirchhoff equation with a logarithmic type perturbation in dimension 4

Critical exponent for global solutions in a fourth‐order hyperbolic equation of p‐Kirchhoff type

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