Blow-up of solutions to critical semilinear wave equations with variable coefficients

Abstract

We verify the critical case p = p 0 (n) of Strauss' conjecture [31] concerning the blow-up of solutions to semilinear wave equations with variable coefficients in R n , where n ≥ 2. The perturbations of Laplace operator are assumed to be smooth and decay exponentially fast at infinity. We also obtain a sharp lifespan upper bound for solutions with compactly supported data when p = p 0 (n). The unified approach to blow-up problems in all dimensions combines several classical ideas in order to generalize and simplify the method of Zhou [44] and Zhou & Han [46]: exponential "eigenfunctions" of the Laplacian [38] are used to construct the test function φq for linear wave equation with variable coefficients and John's method of iterations [14] is augmented with the "slicing method" of Agemi, Kurokawa and Takamura [1] for lower bounds in the critical case.

Nonexistence of Global Solutions for a Weakly Coupled System of Semilinear Damped Wave Equations of Derivative Type in the Scattering Case

“…for any test functions φ, ψ ∈ C ∞ 0 ([0, T ) × R n ) and any t ∈ [0, T ). The remaining part of this section is organized as follows: first, in Section 6.1 we recall some auxiliary functions from [47] and we use them to introduce the functionals for the critical case; in Section 6.2 we derive the iteration frame for these functionals, that is, a coupled system of nonlinear ordinary integral inequalities; in Section 6.3 lower bound estimates for the functionals, that allow to start with the iteration procedure, are derived; then, in Section 6.4 we combine the iteration frame from Section 6.2 and the lower bounds from Section 6.3 with a slicing method; hence, in Section 6.5 we use the sequences of lower bounds for the functionals from Section 6.4 to prove the blow-up result and to establish the upper bound for the lifespan; finally, in Section 6.6 we compare our results with those proved in Section 9 of [16] and we provide the analytic expression of the coordinates of the cusp point for the critical curve in the pq plane.…”

“…Remark 6.4. Let us stress that differently from [47,Lemma 3.1] we require in the statement of (i) and (ii) the condition of r > −1 instead of r > 0. Nonetheless, the proofs from [47] of (i) and of the lower bound for η r (t, s, x) in (ii) are still valid even for r > −1.…”

“…where r > −1, λ 0 is a fixed positive constant and Φ is defined by (27). These auxiliary functions have been introduced in [47] as generalizations of the test function considered by Zhou (see [57, equation (3.2)]) in the treatment of the critical case for the semilinear wave equation with power nonlinearity in the higher dimensional case. Let us underline that the assumption on r is done in order to guarantee the integrability of the function λ r in a neighborhood of 0.…”

Blow-up for a weakly coupled system of semilinear damped wave equations in the scattering case with power nonlinearities

“…The remaining part of this paper is organized as follows: in Section 2 we recall a multiplier, that has been introduced in [18] in order to study the corresponding single semilinear equation, and its properties and we derive some lower bounds for certain functionals related to a local solution; then, in Section 3 we prove Theorem 1.2 by using the preparatory results from Section 2 and an iterative method. Finally, in Section 4 we prove the result in the critical case adapting the approach from [35,36] for a weakly coupled system. In particular, the slicing method is employed in order to deal with logarithmic factors in the iteration argument.…”

Finite time blowup of solutions to semilinear wave equation in an exterior domain

“…This argument requires the special solutions of corresponding linear wave equation having slowly decaying property. To construct this kind of solution, we used the construction by Wakasa-Yordanov [15].…”