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

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