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Existence of global solutions to the Cauchy problem for the semilinear dissipative wave equations. (English) Zbl 0790.35072

We consider the Cauchy problem for the semilinear wave equation with a dissipative term:

u tt -Δu+u t -|u| p u=0in N ×[0,),u(x,0)=u 0 (x),u t (x,0)=u 1 (x)·(*)

By use of “potential well” or “modified potential well” method, we prove the global existence and decay property of weak solutions for the equation (*) with 4/Np<4/(N-2) under the assumption that u 1 2 +u 0 2 is small. For our results, the dissipative term u t plays an essential role. Moreover, by applying the above method, we discuss on the life span of weak solutions for both of the equations with and without dissipative term.

Reviewer: K.Ono (Fukuoka)
35L70Nonlinear second-order hyperbolic equations
35D05Existence of generalized solutions of PDE (MSC2000)
35B40Asymptotic behavior of solutions of PDE
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