A shift in the Strauss exponent for semilinear wave equations with a not effective damping

作者：D'Abbicco Marcello; Lucente Sandra; Reissig Michael^*

来源：Journal of Differential Equations, 2015, 259(10): 5040-5073.

DOI：10.1016/j.jde.2015.06.018

• Semilinear damped wave equation
• Not effective damping
• Small data global existence
• Strauss exponent

In this note we study the global existence of small data solutions to the Cauchy problem for the semilinear wave equation with a not effective scale-invariant damping term, namely
v(tt) - Delta v +2/1+t v(t) = broken vertical bar v broken vertical bar(P), v(0, x) = v(0)(x), v(t) (0, = v(1)(x),
where p > 1, n > 2. We prove blow-up in finite time in the subcritical range p is an element of (1, p(2)(n)] and existence theorems for p > p2(n), n = 2, 3. In this way we find the critical exponent for small data solutions to this problem. Our results lead to the conjecture p2(n) = p(0)(n +2) for n > 2, where p0(n) is the Strauss exponent for the classical semilinear wave equation with power nonlinearity.

• 出版日期2015-11-15