Critical exponent for semi-linear wave equations with double damping terms in exterior domains

  • Marcello D’Abbicco
  • Ryo IkehataEmail author
  • Hiroshi Takeda


In this paper, we consider wave equations with double damping terms expressed by \(u_{t}\) and \(-\Delta u_{t}\) and a power type of nonlinearity \(\vert u\vert ^{p}\). We are concerned with mixed problems for these equations in exterior domains of a bounded obstacle. A main purpose is to determine a so-called critical exponent of the power p of the nonlinearity \(\vert u\vert ^{p}\). In particular, in the two dimensional case, our results are optimal, and the critical exponent is given by the Fujita one. This shows a parabolic aspect (as \(t \rightarrow \infty \)) of our equations considered in exterior domains, and one can see that the usual frictional damping \(u_{t}\) is more dominant than the strong one \(-\Delta u_{t}\) as \(t \rightarrow \infty \) even in the nonlinear problem case.


Wave equation Frictional term Structural damping Power nonlinearities Mixed problem Small data global existence Blowup Fujita exponent 

Mathematics Subject Classification

Primary 35A01 35L05 Secondary 35B40 35B33 35B44 



The work of the second author (R. Ikehata) was supported in part by Grant-in-Aid for Scientific Research (C)15K04958 of JSPS. The work of the third author (H. Takeda) was supported in part by Grant-in-Aid for Young Scientists (B)15K17581 of JSPS. The authors would like to thank the referee for his useful comments.


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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Marcello D’Abbicco
    • 1
  • Ryo Ikehata
    • 2
    Email author
  • Hiroshi Takeda
    • 3
  1. 1.Department of MathematicsUniversity of BariBariItaly
  2. 2.Department of Mathematics, Graduate School of EducationHiroshima UniversityHigashi-HiroshimaJapan
  3. 3.Department of Intelligent Mechanical Engineering, Faculty of EngineeringFukuoka Institute of TechnologyFukuokaJapan

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