The influence of data regularity in the critical exponent for a class of semilinear evolution equations
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Nonlinear Differential Equations and Applications NoDEA
The influence of data regularity in the critical exponent for a class of semilinear evolution equations Marcelo R. Ebert , Cleverson R. da Luz and Ma´ıra F. G. Palma Abstract. In this paper we find the critical exponent for the global existence (in time) of small data solutions to the Cauchy problem for the semilinear dissipative evolution equations utt + (−Δ)δ utt + (−Δ)α u + (−Δ)θ ut = |ut |p ,
t ≥ 0, x ∈ Rn ,
with p > 1, additional 2θ ∈ [0, α] and δ ∈ (θ,α]. We show that, under regularity H α+δ (Rn ) ∩ Lm (Rn ) × H 2δ (Rn ) ∩ Lm (Rn ) for initial data, . The with m ∈ (1, 2], the critical exponent is given by pc = 1 + 2mθ n nonexistence of global solutions in the subcritical cases is proved, in the case of integers parameters α, δ, θ, by using the test function method (under suitable sign assumptions on the initial data). Mathematics Subject Classification. Primary 35B33, 35B40; Secondary 35L71, 35L90. Keywords. Semilinear evolution operators, Structural dissipation, Global small data solutions, Critical exponent, Asymptotic behavior of solutions.
1. Introduction Let us consider the Cauchy problem for the semilinear dissipative evolution equations utt + (−Δ)δ utt + (−Δ)α u + (−Δ)θ ut = |ut |p , t ≥ 0, x ∈ Rn , (1.1) (u, ut )(0, x) = (u0 , u1 )(x), The first author have been partially supported by Funda¸c˜ ao de Amparo ` a Pesquisa do Estado de S˜ ao Paulo (FAPESP), Grant Number 2017/19497-3. The second author has been partially supported by Conselho Nacional de Desenvolvimento Cient´ıfico e Tecnol´ ogico CNPq, Proc. 308868/2015-3 and 314398/2018-0. 0123456789().: V,-vol
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M. R. Ebert, C. R. da Luz, and M. F. G. Palma
NoDEA b
with p > 1, 2θ ∈ [0, α] and δ ∈ [0, α]. Here we denote by (−Δ) 2 = |D|b , with b ≥ 0, the fractional Laplacian operator defined by its action |D|b f = b F−1 (|ξ| fˆ), where F is the Fourier transform with respect to the space variable, and fˆ = Ff . The case α = 2 and δ = 0 in (1.1) is an important model in the literature, it is known as Germain–Lagrange operator, as well as beam operator and plate operator in the case of space dimension n = 1 and n = 2, respectively. Models to study the vibrations of thin plates given by the full von K´ arm´an system have been studied by several authors, in particular, see [3,20]. If δ = 1 in (1.1), the term −Δutt is to absorb the effects of rotational inertia on the system at the point x of the plate in a positive time t. For the plate equation with exterior damping utt − Δutt + (−Δ)2 u + ut = f (u, ut ), t ≥ 0, x ∈ Rn , (1.2) (u, ut )(0, x) = (u0 , u1 )(x), where f (u, ut ) = |∂tj u|p , j = 0, 1, we address the reader to [1,4,18,24] for a detailed investigation of properties like existence, uniqueness, energy estimates for the solution and global existence (in time) of small data solutions. The derived estimates in Sect. 4 for solutions to the associated linear problem to (1.1) could also be applied to generalize the obtained results in [4], namely, problem (1.2) with power n
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