LAUSR

Abstract We study the Cauchy problem for the Euler-Poisson-Darboux equation, with a power nonlinearity: u t t − u x x + μ t u t = t α | u | p , t > t 0 , x ∈ R , where μ > 0 , p > 1 and α > − 2 . Here either t 0 = 0 (singular problem) or t 0 > 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 , μ 2 + 1 p } > 2 + α . In the case of α = 0 , the above condition is equivalent to p > p crit = max ⁡ { p Str ( 1 + μ ) , 3 } , where p Str ( 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 p crit = ∞ (for μ = 0 ) to p crit = 3 (for μ ≥ 4 / 3 ). The optimality of p crit follows by known nonexistence counterpart results for 1 p ≤ p crit (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.