Asymptotic Analysis

In this article, we investigate Gevrey and summability properties of the formal power series solutions of the inhomogeneous generalized Boussinesq equations. Even if the case that really matters physically is an analytic inhomogeneity, we systematically examine here the cases where the inhomogeneity is s-Gevrey for any s ⩾ 0 , in order to carefully distinguish the influence of the data (and their degree of regularity) from that of the equation (and its structure). We thus prove that we have a noteworthy dichotomy: for any s ⩾ 1 , the formal solutions and the inhomogeneity are simultaneously s-Gevrey; for any s < 1 , the formal solutions are generically 1-Gevrey. In the latter case, we give in particular an explicit example in which the formal solution is s ′ -Gevrey for no s ′ < 1 , that is exactly 1-Gevrey. Then, we give a necessary and sufficient condition under which the formal solutions are 1-summable in a given direction arg ( t ) = θ . In addition, we present some technical results on the generalized binomial and multinomial coefficients, which are needed for the proofs of our various results.

The chemotaxis system ( ⋆ ) u t = ∇ · ( D ( u ) ∇ u ) − ∇ · ( u S ( u ) ∇ v ) , 0 = Δ v − μ + u , μ = 1 | Ω | ∫ Ω u , is considered in a ball Ω = B R ( 0 ) ⊂ R n .

It is shown that if S ∈ C 2 ( [ 0 , ∞ ) ) suitably generalizes the prototype given by S ( ξ ) = χ ξ + 1 , ξ ⩾ 0 , with some χ > 0 , and if diffusion is suitably weak in the sense that 0 < D ∈ C 2 ( ( 0 , ∞ ) ) is such that there exist K D > 0 and m ∈ ( − ∞ , 1 − 2 n ) fulfilling D ( ξ ) ⩽ K D ξ m − 1 for all  ξ > 0 , then for appropriate choices of sufficiently concentrated initial data, an associated no-flux initial-boundary value problem admits a global classical solution ( u , v ) which blows up in infinite time and satisfies 1 C e χ t ⩽ ‖ u ( · , t ) ‖ L ∞ ( Ω ) ⩽ C e χ t for all  t > 0 . A major part of the proof is based on a comparison argument involving explicitly constructed subsolutions to a scalar parabolic problem satisfied by mass accumulation functions corresponding to solutions of (⋆).

We present the two-scale-transformation method which allows rigorous homogenisation of problems defined in locally periodic domains. This method transforms such problems into periodic domains in order to facilitate the passage to the limit. The idea of transforming problems into periodic domains originates from the homogenisation of problems defined in evolving microstructure and has been applied in several works. However, only the homogenisation of the periodic substitute problems was proven, whereas the method itself was just postulated (i.e. the equivalence to the homogenisation of the actual problem had to be assumed). In this work, we develop this idea further and formulate a rigorous two-scale convergence concept for microscopic transformation. Thus, we can prove that the homogenisation of the periodic substitute problem is equivalent to the homogenisation of the actual problem. Moreover, we show a new two-scale transformation rule for gradients which enables the derivation of new limit problems that are now transformationally independent.

We consider the asymptotic behaviour of small-amplitude gravity water waves in a rectangular domain where the water depth is much smaller than the horizontal scale. The control acts on one lateral boundary, by imposing the horizontal acceleration of the water along that boundary, as a scalar input function u. The state z of the system consists of two functions: the water level ζ along the top boundary, and its time derivative ∂ ζ ∂ t . We prove that the solution of the water waves system converges to the solution of the one dimensional wave equation with Neumann boundary control, when taking the shallowness limit. Our approach is based on a special change of variables and a scattering semigroup, which provide the possiblity to apply the Trotter–Kato approximation theorem. Moreover, we use a detailed analysis of Fourier series for the dimensionless version of the partial Dirichlet to Neumann and Neumann to Neumann operators.

In this paper we study the longtime dynamics of a class of thermoelastic Timoshenko beams with history in a nonlinear elastic foundation. Our main result establishes the existence of a global attractor with finite fractal dimension without requiring the so-called equal wave speeds assumption. In addition, the attractor belongs to the phase space of strong solutions. The results are based on properties of gradient systems and a concept of quasi-stability. We believe this is the first study on the existence of global attractors for semilinear Timoshenko systems with hybrid dissipation (heat and memory).

The objective of this work is to investigate a nonlocal problem involving singular and critical nonlinearities: ( [ u ] s , p p ) σ − 1 ( − Δ ) p s u = λ u γ + u p s ∗ − 1 in  Ω , u > 0 , in  Ω , u = 0 , in  R N ∖ Ω , where Ω is a bounded domain in R N with the smooth boundary ∂ Ω , 0 < s < 1 < p < ∞ , N > s p , 1 < σ < p s ∗ / p , with p s ∗ = N p N − p s , ( − Δ ) p s is the nonlocal p-Laplace operator and [ u ] s , p is the Gagliardo p-seminorm. We combine some variational techniques with a truncation argument in order to show the existence and the multiplicity of positive solutions to the above problem.