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Behavior of solutions to a Petrovsky equation with damping and variable-exponent sources

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Abstract

This paper deals with the following Petrovsky equation with damping and nonlinear sources:

$${u_{tt}} + {\Delta ^2}u - M\left({\left\| {\nabla u} \right\|_2^2} \right)\Delta u - \Delta {u_t} + {\left| {{u_t}} \right|^{m(x) - 2}}{u_t} = {\left| u \right|^{p(x) - 2}}u$$

under initial-boundary value conditions, where M(s) = a + bsγ is a positive C1 function with the parameters a > 0, b > 0, γ ⩾ 1, and m(x) and p(x) are given measurable functions. The upper bound of the blow-up time is derived for low initial energy by the differential inequality technique. For m(x) ≡ 2, in particular, the upper bound of the blow-up time is obtained by the combination of Levine’s concavity method and some differential inequalities under high initial energy. In addition, we discuss the lower bound of the blow-up time by making full use of the strong damping. Moreover, we present the global existence of solutions and an energy decay estimate by establishing some energy estimates.

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Acknowledgements

This work was supported by National Natural Science Foundation of China (Grant No. 12071391). The first author expresses her gratitude to Professor Wenjie Gao and Bin Guo in School of Mathematics, Jilin University for their support and constant encouragement. In particular, the authors thank Professor Baisheng Yan in Department of Mathematics, Michigan State University for improving the quality of this paper. The authors are grateful to the anonymous referees for their careful reading of the manuscript and many helpful suggestions.

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  1. School of Mathematical Sciences, Xiamen University, Xiamen, 361005, China

    Menglan Liao & Zhong Tan

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  1. Menglan Liao
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  2. Zhong Tan
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Correspondence to Zhong Tan.

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Liao, M., Tan, Z. Behavior of solutions to a Petrovsky equation with damping and variable-exponent sources. Sci. China Math. 66, 285–302 (2023). https://doi.org/10.1007/s11425-021-1926-x

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