We study the Cauchy problem for the semilinear structural damped wave equation with source term uttMathClass-bin−MathClass-rel△uMathClass-bin+μ(MathClass-bin−Δ)σutMathClass-rel=f(u)MathClass-punc,1emquadu(0MathClass-punc,x)MathClass-rel=u0(x)MathClass-punc,1emquadut(0MathClass-punc,x)MathClass-rel=u1(x)MathClass-punc,with σ ∈ (0,1] in space dimension n ≥ 2 and with a positive constant μ. We are interested in the influence of σ on the critical exponent pcrit in | f(u) | ≈ | u | p. This critical exponent is the threshold between global existence in time of small data solutions and blow‐up behavior for some suitable range of p. Our results are optimal for σ = 1 ∕ 2. Copyright © 2013 John Wiley & Sons, Ltd.

Abstract. In this note we study the global existence of small data solutions to the Cauchy problem for the semi-linear wave equation with a not effective scale-invariant damping term, namelywhere p > 1, n ≥ 2. We prove blow-up in finite time in the subcritical range p ∈ (1, p 2 (n)] and an existence result for p > p 2 (n), n = 2, 3. In this way we find the critical exponent for small data solutions to this problem. All these considerations lead to the conjecture p 2 (n) = p 0 (n + 2) for n ≥ 2, where p 0 (n) is the Strauss exponent for the classical wave equation.

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The authors study the Cauchy problem for the semi-linear damped wave equation\ud in any space dimension. It is assumed that the time-dependent damping term is effective. The global existence of small energy\ud data solutions for polynomial nonlinear term in the supercritical case is proved.The authors study the Cauchy problem for the semi-linear damped wave equation utt - Δu + b(t)ut = f(u), u(0,x) =u0(x), ut(0,x) = u1(x) in any space dimension n ≥ 1. It is assumed that the time-dependent damping term b(t) > 0 is effective, and in particular tb(t) → ∞ as t → ∞. The global existence of small energy data solutions for pipef(u)pipe ≈ pipeupipep in the supercritical case of p > 1 + 2/n and p ≤ n/n-2 for n ≥ 3 is proved. © 2013 Fudan University and Springer-Verlag Berlin Heidelberg