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Global existence for semi-linear structurally damped \(\sigma\)-evolution models. (English) Zbl 1327.35411

Summary: The main purpose of this paper is to study the global existence of small data solutions for semi-linear structurally damped \(\sigma\)-evolution models of the form \[ u_{t t} +(- {\Delta})^\sigma u + \mu(- {\Delta})^\delta u_t = f(| D |^a u, u_t), u(0, x) = u_0(x), u_t(0, x) = u_1(x) \] with \(\sigma \geq 1\), \(\mu > 0\) and \(\delta = \frac{\sigma}{2}\). This is a family of structurally damped \(\sigma\)-evolution models interpolating between models with exterior damping \(\delta = 0\) and those with visco-elastic type damping \(\delta = \sigma\). The function \(f(| D |^a u, u_t)\) represents power non-linearities \(| | D |^a u |^p\) for \(a \in [0, \sigma)\) or \(| u_t |^p\). Our goal is to propose a Fujita type exponent diving the admissible range of powers \(p\) into those allowing global existence of small data solutions (stability of zero solution) and those producing a blow-up behavior even for small data. On the one hand we use new results from harmonic analysis for fractional Gagliardo-Nirenberg inequality or for superposition operators (see Appendix A), on the other hand our approach bases on \(L^p - L^q\) estimates not necessarily on the conjugate line for solutions to the corresponding linear models assuming additional \(L^1\) regularity for the data. The linear models we have here in mind are \[ v_{t t} +(- {\Delta})^\sigma v + \mu(- {\Delta})^\delta v_t = 0, v(0, x) = v_0(x), v_t(0, x) = v_1(x) \] with \(\sigma \geq 1\), \(\mu > 0\) and \(\delta \in(0, \sigma]\).

MSC:

35R11 Fractional partial differential equations
26A33 Fractional derivatives and integrals
35A01 Existence problems for PDEs: global existence, local existence, non-existence
35B33 Critical exponents in context of PDEs
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