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

Abstract

In this paper, we establish blow-up results for the semilinear wave equation in generalized Einstein-de Sitter spacetime with nonlinearity of derivative type. Our approach is based on the integral representation formula for the solution to the corresponding linear problem in the onedimensional case, that we will determine through Yagdjian's Integral Transform approach. As upper bound for the exponent of the nonlinear term, we discover a Glassey-type exponent which depends both on the space dimension and on the Lorentzian metric in the generalized Einstein-de Sitter spacetime.

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

“…where θ(n, ℓ, µ, p) is defined in (13) and is a positive quantity thanks to the condition p < p Str (n+ µ ℓ+1 , ℓ) on the exponent of the nonlinear term. Also, for j j 0 and t 2T 1 we arrived at…”

“…Moreover, under the same conditions for the parameters as above, also the case with derivative type nonlinearity |∂ t u| p is studied in [13,14,36]. Furthermore, we point out that in [35,36] even the case ℓ −1 with ν 2 = 0 is studied for different semilinear terms.…”

Fundamental solutions for the Dirac equation in curved spacetime and generalized Euler-Poisson-Darboux equation

“…where the operators A k (x, ∂ x ), k = 1, 2, 3, in general, are the scalar pseudo-differential operators A k (x, ∂ x ) with symbols depending on the spatial variables too. Following [33] we define the generalized Dirac operator as an operator (14) satisfying the condition…”

“…The complementary operator D co (x, t, ∂ t , ∂ x ) for operator (14) is given by the following theorem.…”

“…In [21] the version of integral transform approach [30,32] was employed by Palmieri for the derivation of the integral representation formulas for the solution of a linear equation with the coefficients, which are scale-invariant and independent of x variable. Then Palmieri and coauthors [14] applied these formulas to examine the blowup phenomena for the wave equations with different types of nonlinearities in spacetime with power type expansion.…”