A new phenomenon in the critical exponent for structurally damped semi-linear evolution equations. (English) Zbl 1355.35015
Summary: In this paper, we find the critical exponent for global small data solutions to the Cauchy problem in \(\mathbb{R}^n\), for dissipative evolution equations with power nonlinearities \(| u |^p\) or \(| u_t |^p\),
\[
u_{t t} +(- \Delta)^\delta u_t +(- \Delta)^\sigma u = \begin{cases} | u |^p, \\ | u_t |^p. \end{cases}
\]
Here \(\sigma, \delta \in \mathbb{N} \setminus \{0 \}\), with \(2 \delta \leq \sigma\). We show that the critical exponent for each of the two nonlinearities is related to each of the two possible asymptotic profiles of the linear part of the equation, which are described by the diffusion equations:
\[
v_t +(- \Delta)^{\sigma - \delta} v = 0, w_t +(- \Delta)^\delta w = 0.
\]
The nonexistence of global solutions in the critical and subcritical cases is proved by using the test function method (under suitable sign assumptions on the initial data), and lifespan estimates are obtained. By assuming small initial data in Sobolev spaces, we prove the existence of global solutions in the supercritical case, up to some maximum space dimension \(\overline{n}\), and we derive \(L^q\) estimates for the solution, for \(q \in(1, \infty)\). For \(\sigma = 2 \delta\), the result holds in any space dimension \(n \geq 1\). The existence result also remains valid if \(\sigma\) and/or \(\delta\) are fractional.
MSC:
| 35B33 | Critical exponents in context of PDEs |
| 35R11 | Fractional partial differential equations |
| 35L15 | Initial value problems for second-order hyperbolic equations |
| 35L30 | Initial value problems for higher-order hyperbolic equations |
| 35L71 | Second-order semilinear hyperbolic equations |
| 35L76 | Higher-order semilinear hyperbolic equations |
| 35L82 | Pseudohyperbolic equations |
| 35B40 | Asymptotic behavior of solutions to PDEs |
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