Small data solutions for the Euler-Poisson-Darboux equation with a power nonlinearity

Abstract

We study the Cauchy problem for the Euler-Poisson-Darboux equation, with a power nonlinearity:utt−uxx+μtut=tα|u|p,t>t0,x∈R, where μ>0, p>1 and α>−2. Here either t0=0 (singular problem) or t0>0 (regular problem). We show that this model may be interpreted as a semilinear wave equation with borderline dissipation: the existence of global small data solutions depends not only on the power p, but also on the parameter μ. Global small data weak solutions exist if(p−1)min⁡{1,μ,μ2+1p}>2+α. In the case of α=0, the above condition is equivalent to p>pcrit=max⁡{pStr(1+μ),3}, where pStr(k) is the critical exponent conjectured by W.A. Strauss for the semilinear wave equation without dissipation (i.e. μ=0) in space dimension k. Varying the parameter μ, there is a continuous transition from pcrit=∞ (for μ=0) to pcrit=3 (for μ≥4/3). The optimality of pcrit follows by known nonexistence counterpart results for 1<p≤pcrit (and for any p>1 if μ=0).As a corollary of our result, we obtain analogous results for generalized semilinear Tricomi equations and other models related to the Euler-Poisson-Darboux equation.