In this article, the homogenization process of periodic structures is analyzed using Bloch waves in the case of system of linear elasticity in three dimensions. The Bloch wave method for homogenization relies on the regularity of the lower Bloch spectrum. For the three dimensional linear elasticity system, the first eigenvalue is degenerate of multiplicity three and hence existence of such a regular Bloch spectrum is not guaranteed. The aim here is to develop all necessary spectral tools to overcome...

In this article, the homogenization process of periodic structures is analyzed using Bloch waves in the case of system of linear elasticity in three dimensions. The Bloch wave method for homogenization relies on the regularity of the lower Bloch spectrum. For the three dimensional linear elasticity system, the first eigenvalue is degenerate of multiplicity three and hence existence of such a regular Bloch spectrum is not guaranteed. The aim here is to develop all necessary spectral tools to overcome...

We present the fiber-spring elastic model of the arterial wall with atherosclerotic plaque composed of a lipid pool and a fibrous cap. This model allows us to reproduce pressure to cross-sectional area relationship along the diseased vessel which is used in the network model of global blood circulation. Atherosclerosis attacks a region of systemic arterial network. Our approach allows us to examine the impact of the diseased region onto global haemodynamics....

These notes present the main results of [22, 23, 24] concerning the mass critical (gKdV) equation $u_t+{(u_{xx}+u^5)}_x=0$ for initial data in $H^1$ close to the soliton. These works revisit the blow up phenomenon close to the family of solitons in several directions: definition of the stable blow up and classification of all possible behaviors in a suitable functional setting, description of the minimal mass blow up in $H^1$, construction of various exotic blow up rates in $H^1$, including grow up in infinite time.

We consider the critical nonlinear Schrödinger equation $i{u}_t=-\Delta u-{\left|u\right|}^{\frac{4}{N}}u$ with initial condition $u(0,x)=u_0$ in dimension $N$. For $u_0\in H^1$, local existence in time of solutions on an interval $[0,T)$ is known, and there exists finite time blow up solutions, that is $u_0$ such that ${lim}_{t\to T\<+\infty }{\left|u_x\left(t\right)\right|}_{L^2}=+\infty$. This is the smallest power in the nonlinearity for which blow up occurs, and is critical in this sense. The question we address is to understand the blow up dynamic. Even though there exists an explicit example of blow up solution and a class of initial data...

This paper provides blow up results of Fujita type for a reaction-diffusion system of 3 equations in the form $uₜ-\Delta \left(a_{11}u\right)=h(t,x){\left|v\right|}^p$, $vₜ-\Delta \left(a_{21}u\right)-\Delta \left(a_{22}v\right)=k(t,x){\left|w\right|}^q$, $wₜ-\Delta \left(a_{31}u\right)-\Delta \left(a_{32}v\right)-\Delta \left(a_{33}w\right)=l(t,x){\left|u\right|}^r$, for $x\in {ℝ}^N$, t > 0, p > 0, q > 0, r > 0, $a_{ij}=a_{ij}(t,x,u,v)$, under initial conditions u(0,x) = u₀(x), v(0,x) = v₀(x), w(0,x) = w₀(x) for $x\in {ℝ}^N$, where u₀, v₀, w₀ are nonnegative, continuous and bounded functions. Subject to conditions on dependence on the parameters p, q, r, N and the growth of the functions h, k, l at infinity, we prove finite blow up time for every solution of the above system,...

We consider the mass critical (gKdV) equation $u_t+{(u_{xx}+u^5)}_x=0$ for initial data in $H^1$. We first prove the existence and uniqueness in the energy space of a minimal mass blow up solution and give a sharp description of the corresponding blow up soliton-like bubble. We then show that this solution is the universal attractor of all solutions near the ground state which have a defocusing behavior. This allows us to sharpen the description of near soliton dynamics obtained in [29].

We prove Fujita-type global existence and nonexistence theorems for a system of m equations (m > 1) with different diffusion coefficients, i.e. $u_{it}-d_i\Delta u_i={\prod }_{k=1}^m{u}_k^{p_k^i},i=1,...,m,x\in {ℝ}^N,t>0,$ with nonnegative, bounded, continuous initial values and $p_k^i\ge 0$, $i,k=1,...,m$, $d_i>0$, $i=1,...,m$. For solutions which blow up at $t=T<\le \infty$, we derive the following bounds on the blow up rate: $u_i(x,t)\le C{(T-t)}^{-{\alpha }_i}$ with C > 0 and ${\alpha }_i$ defined in terms of $p_k^i$.