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On the Cauchy problem for semi-linear σ -evolution equations with time-dependent damping
  • Tuan Anh Dao,
  • Halit Sevki Aslan
Tuan Anh Dao
Hanoi University of Science and Technology School of Applied Mathematics and Informatics

Corresponding Author:[email protected]

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Halit Sevki Aslan
Universidade de Sao Paulo Campus de Ribeirao Preto
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In this paper, we would like to consider the Cauchy problem for semi-linear σ-evolution equations with time-dependent damping for any σ≥1. Motivated strongly by the classification of damping terms in the paper34, the first main goal of the present work is to make some generalizations from σ=1 to σ>1 and simultaneously to investigate decay estimates for solutions to the corresponding linear equations in the so-called effective damping cases. For the next main goals, we are going not only to prove the global well-posedness property of small data solutions but also to indicate blow-up results for solutions to the semi-linear problem. In this concern, the novelty which should be recognized is that the application of a modified test function combined with a judicious choice of test functions gives blow-up phenomena and upper bound estimates for lifespan in both the subcritical case and the critical case, where σ is assumed to be any fractional number. Finally, lower bound estimates for lifespan in some spatial dimensions are also established to find out their sharp results.
09 Mar 2023Submitted to Mathematical Methods in the Applied Sciences
09 Mar 2023Submission Checks Completed
09 Mar 2023Assigned to Editor
14 Mar 2023Review(s) Completed, Editorial Evaluation Pending
22 Mar 2023Reviewer(s) Assigned
22 Jul 2023Editorial Decision: Revise Major
11 Aug 20231st Revision Received
11 Aug 2023Submission Checks Completed
11 Aug 2023Assigned to Editor
11 Aug 2023Review(s) Completed, Editorial Evaluation Pending
29 Aug 2023Reviewer(s) Assigned