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

On the the critical exponent for the semilinear Euler-Poisson-Darboux-Tricomi equation with power nonlinearity

“…We point out that the condition δ 0 implies somehow that the damping term µt −1 ∂ t u has a dominant influence over the mass term ν 2 t −2 u. Although the proof of the necessity part of this conjecture is fully demonstrated (see [29,30,22]), for the sufficiency part only the one-dimensional case was recently completely clarified [5], while in the higher dimensional case only a few special cases have been clarified (often in the radially symmetric case).…”

“…For ℓ = 0 and ν 2 = 0, the linearized equation associated with the equation in ( 1) is called the Euler-Poisson-Darboux equation (see the introduction of [5] for a detailed overview on the literature regarding this model), while for µ = ν 2 = 0 the second-order operator ∂ 2 t − t 2ℓ ∆ on the left-hand side of the equation in ( 1) is called generalized Tricomi operator. Motivated by these special cases and for the sake of brevity, we will call the equation in (1) semilinear Euler-Poisson-Darboux-Tricomi equation (semilinear EPDT equation).…”

“…Regarding the semilinear Euler-Poisson-Darboux equation and, more in general, the semilinear wave equation with scale-invariant damping and mass (i.e. (1) for ℓ = 0), combining the contributions from many different authors (see [4,7,25,19,29,30,8,22,5,3] and references therein) we may reasonably conjecture that for nonnegative µ, ν 2 such that the quantity…”

“…Furthermore, we point out that in [35,36] even the case ℓ −1 with ν 2 = 0 is studied for different semilinear terms. The purpose of the present paper is twofold: on the one hand, we want to prove the necessity part for the one-dimensional case in (1) that together with the sufficiency part from [5] (cf. Corollary 5.1) will show the optimality of our result for n = 1; on the other hand, we want to generalize the condition that determines the critical exponent for (1) in the general n-dimensional case, obtaining as a candidate to be critical an exponent that is consistent with all previously exposed special cases.…”

A note on the nonexistence of global solutions to the semilinear wave equation with nonlinearity of derivative-type in the generalized Einstein-de Sitter spacetime

“…has a crucial role in determining some properties of the fundamental solution of L k,µ,ν 2 . In the special case k = 0 (the so-called wave operator with scale-invariant damping and mass), it is known in the literature that the value of δ affects not only the fundamental solution of L 0,µ,ν 2 but also the critical exponents in the treatment of semilinear Cauchy problem associated with L 0,µ,ν 2 with power nonlinearity [11,16,17,1], nonlinearity of derivative type [18], and combined nonlinearity [4,5].…”