Acta mathematica scientia,Series A ›› 2020, Vol. 40 ›› Issue (6): 1568-1589.

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Global Existence and Blowup Phenomena for a Semilinear Wave Equation with Time-Dependent Damping and Mass in Exponentially Weighted Spaces

Changwang Xiao1(),Fei Guo1,2,*()   

  1. 1 School of Mathematical Sciences, Nanjing Normal University, Nanjing 210023
    2 School of Mathematical Sciences and Jiangsu Key Laboratory for NSLSCS, Nanjing Normal University, Nanjing 210023
  • Received:2019-10-25 Online:2020-12-26 Published:2020-12-29
  • Contact: Fei Guo;
  • Supported by:
    Priority Academic Program Development of Jiangsu Higher Education Institutions, the NSF of Jiangsu Province(BK20181381);the NSF of the Jiangsu Higher Education Institutions(17KJA110002);the Qing Lan Projects of Jiangsu Province


We consider the global small data solutions and blowup to the Cauchy problem for a semilinear wave equation with time-dependent damping and mass term as well as power nonlinearity. On one hand, if the power of the nonlinearity $p >p_F(N)=1+ \frac 2N$, it is proved that solutions with small initial data exist for all time in exponentially weighted energy spaces. On the other hand, if the power satisfies <p\leq p_F(\alpha, n)=1+\frac{2(1+\alpha)}{N(1+\alpha)-2\alpha}~(0<\alpha<1)$, for some special chosen parameters it is shown that solutions must blow up in finite time provided that the initial data satisfy some integral sign conditions.

Key words: Semilinear wave equation, Global solution, Blow up, Fujita exponent

CLC Number: 

  • O175.27