A pneumatic tyre in rotating motion with a constant angular velocity $\Omega$ is assimilated to a torus whose generating circle has a radius $R$. The contact of the tyre with the ground is schematized as an ellipse with semi-major axis $a$. When $(\Omega R/C_0)\ll 1$ and $(a/R)\ll 1$ (where $C_0$ is the velocity of the sound), we show that at the rapid time scale $R/C_0$, the air motion within a torus periodically excited on its surface generates an acoustic wave $h$. A study of this acoustic wave is conducted and shows that the mode associated...

Large time behavior of solutions to the generalized damped wave equation $u_{tt}+A{u}_t+\nu Bu+F(x,t,u,u_t,\nabla u)=0$ for $(x,t)\in {ℝ}^n\times [0,\infty )$ is studied. First, we consider the linear nonhomogeneous equation, i.e. with F = F(x,t) independent of u. We impose conditions on the operators A and B, on F, as well as on the initial data which lead to the selfsimilar large time asymptotics of solutions. Next, this abstract result is applied to the equation where $A{u}_t=u_t$, $Bu=-\Delta u$, and the nonlinear term is either $|u_t{|}^{q-1}{u}_t$ or ${\left|u\right|}^{\alpha -1}u$. In this case, the asymptotic profile of solutions...

The nonlinear dissipative wave equation $u_{tt}-\Delta u+{\left|u_t\right|}^{h-1}{u}_t=0$ in dimension $d\>1$ has strong solutions with the following structure. In $0\le t\<1$ the solutions have a focusing wave of singularity on the incoming light cone $\left|x\right|=1-t$. In $\{t\ge 1\}$ that is after the focusing time, they are smoother than they were in $\{0\le t\<1\}$. The examples are radial and piecewise smooth in $\{0\le t\<1\}$

We study the statistical properties of the solutions of the Kadomstev-Petviashvili equations (KP-I and KP-II) on the torus when the initial datum is a random variable. We give ourselves a random variable $u_0$ with values in the Sobolev space $H^s$ with $s$ big enough such that its Fourier coefficients are independent from each other. We assume that the laws of these Fourier coefficients are invariant under multiplication by $e^{i\theta }$ for all $\theta \in ℝ$. We investigate about the persistence of the decorrelation...

Let WF⁎ be the wave front set with respect to $C^{\infty }$, quasi analyticity or analyticity, and let K be the kernel of a positive operator from $C{₀}^{\infty }$ to ’. We prove that if ξ ≠ 0 and (x,x,ξ,-ξ) ∉ WF⁎(K), then (x,y,ξ,-η) ∉ WF⁎(K) and (y,x,η,-ξ) ∉ WF⁎(K) for any y,η. We apply this property to positive elements with respect to the weighted convolution $u{\ast }_B\phi \left(x\right)=\int u(x-y)\phi \left(y\right)B(x,y)dy$, where $B\in C^{\infty }$ is appropriate, and prove that if $(u{\ast }_B\phi ,\phi )\ge 0$ for every $\phi \in C{₀}^{\infty }$ and (0,ξ) ∉ WF⁎(u), then (x,ξ) ∉ WF⁎(u) for any x.

The semilinear differential equation (1), (2), (3), in $ℝ\times \mathrm{\Omega }$ with $\mathrm{\Omega }\in {ℝ}^N$, (a nonlinear wave equation) is studied. In particular for $\mathrm{\Omega }={ℝ}^3$, the existence is shown of a weak solution $u(t,x)$, periodic with period $T$, non-constant with respect to $t$, and radially symmetric in the spatial variables, that is of the form $u(t,x)=\nu (t,|x|)$. The proof is based on a distributional interpretation for a linear equation corresponding to the given problem, on the Paley-Wiener criterion for the Laplace Transform, and on the alternative...

We investigate for which metric $g$ (close to the standard metric $g_0$) the solutions of the corresponding d’Alembertian behave like free solutions of the standard wave equation. We give rather weak (, non integrable) decay conditions on $g-g_0$; in particular, $g-g_0$ decays like $t^{-\frac{1}{2}-\epsilon }$ along wave cones.

We consider solutions of quasilinear equations $u_t=\Delta u^m+u^p$ in ${ℝ}^N$ with the initial data $u_0$ satisfying $0<u_0<M$ and ${lim}_{\left|x\right|\to \infty }{u}_0\left(x\right)=M$ for some constant $M>0$. It is known that if $0<m<p$ with $p>1$, the blow-up set is empty. We find solutions $u$ that blow up throughout ${ℝ}^N$ when $m>p>1$.

Let $\Omega$ be a bounded domain of class $C^2$ in $ℝ$N and let $K$ be a compact subset of $\partial \Omega$. Assume that $q\ge (N+1)/\left(N-1\right)$ and denote by $U_K$ the maximal solution of $-\Delta u+u^q=0$ in $\Omega$ which vanishes on $\partial \Omega \setminus K$. We obtain sharp upper and lower estimates for $U_K$ in terms of the Bessel capacity $C_{2/q,q^{\text{'}}}$ and prove that $U_K$ is $\sigma$-moderate. In addition we describe the precise asymptotic behavior of $U_K$ at points $\sigma \in K$, which depends on the “density” of $K$ at $\sigma$, measured in terms of the capacity $C_{2/q,q^{\text{'}}}$.

Given a measurable set $A\subset {ℝ}^n$ of positive measure, it is not difficult to show that $|A+A|=\left|2A\right|$ if and only if $A$ is equal to its convex hull minus a set of measure zero. We investigate the stability of this statement: If $\left(\right|A+A|-|2A\left|\right)/\left|A\right|$ is small, is $A$ close to its convex hull? Our main result is an explicit control, in arbitrary dimension, on the measure of the difference between $A$ and its convex hull in terms of $\left(\right|A+A|-|2A\left|\right)/\left|A\right|$.

The semilinear differential equation (1), (2), (3), in $ℝ\times \mathrm{\Omega }$ with $\mathrm{\Omega }\in {ℝ}^N$, (a nonlinear wave equation) is studied. In particular for $\mathrm{\Omega }={ℝ}^3$, the existence is shown of a weak solution $u(t,x)$, periodic with period $T$, non-constant with respect to $t$, and radially symmetric in the spatial variables, that is of the form $u(t,x)=\nu (t,|x|)$. The proof is based on a distributional interpretation for a linear equation corresponding to the given problem, on the Paley-Wiener criterion for the Laplace Transform, and on the alternative...

In this note, we discuss the nonlinear stability in exponential time of Minkowski space-time with a translation space-like Killing field, proved in [13]. In the presence of such a symmetry, the $3+1$ vacuum Einstein equations reduce to the $2+1$ Einstein equations with a scalar field. We work in generalized wave coordinates. In this gauge Einstein equations can be written as a system of quasilinear quadratic wave equations. The main difficulty in [13] is due to the decay in $1/\sqrt{t}$ of free solutions...

We consider the equation $\mathrm{d}y\left(t\right)/\mathrm{d}t=(A+B(t\left)\right)y\left(t\right)$ $(t\ge 0)$, where $A$ is the generator of an analytic semigroup ${\left({\mathrm{e}}^{At}\right)}_{t\ge 0}$ on a Banach space $𝒳$, $B\left(t\right)$ is a variable bounded operator in $𝒳$. It is assumed that the commutator $K\left(t\right)=AB\left(t\right)-B\left(t\right)A$ has the following property: there is a linear operator $S$ having a bounded left-inverse operator $S_l^{-1}$ such that $\parallel S{\mathrm{e}}^{At}\parallel$ is integrable and the operator $K\left(t\right)S_l^{-1}$ is bounded. Under these conditions an exponential stability test is derived. As an example we consider a coupled system of parabolic equations. ...

We study the chemotaxis system with singular sensitivity and logistic-type source: $u_t=\Delta u-\chi \nabla ·(u\nabla v/v)+ru-\mu u^k$, $0=\Delta v-v+u$ under the non-flux boundary conditions in a smooth bounded domain $\Omega \subset {ℝ}^n$, $\chi ,r,\mu >0$, $k>1$ and $n\ge 1$. It is shown with $k\in (1,2)$ that the system possesses a global generalized solution for $n\ge 2$ which is bounded when $\chi >0$ is suitably small related to $r>0$ and the initial datum is properly small, and a global bounded classical solution for $n=1$.

We consider the equation $$-y^{\text{'}}\left(x\right)+q\left(x\right)y(x-\varphi \left(x\right))=f\left(x\right),x\in ℝ,$$ where $\varphi$ and $q$ ($q\ge 1$) are positive continuous functions for all $x\in ℝ$ and $f\in C\left(ℝ\right)$. By a solution of the equation we mean any function $y$, continuously differentiable everywhere in $ℝ$, which satisfies the equation for all $x\in ℝ$. We show that under certain additional conditions on the functions $\varphi$ and $q$, the above equation has a unique solution $y$, satisfying the inequality $$\parallel y^{\text{'}}{\parallel }_{C\left(ℝ\right)}+{\parallel qy\parallel }_{C\left(ℝ\right)}\le c{\parallel f\parallel }_{C\left(ℝ\right)},$$ where the constant $c\in (0,\infty )$ does not depend on the choice of $f$.

We consider the systems of hyperbolic equations ⎧$uₜₜ=\Delta \left(a(t,x)u\right)+\Delta \left(b(t,x)v\right)+{h(t,x)\left|v\right|}^p$, t > 0, $x\in {ℝ}^N$, (S1) ⎨ ⎩$vₜₜ=\Delta \left(c(t,x)v\right)+{k(t,x)\left|u\right|}^q$, t > 0, $x\in {ℝ}^N$ ⎧$uₜₜ=\Delta \left(a(t,x)u\right)+{h(t,x)\left|v\right|}^p$, t > 0, $x\in {ℝ}^N$, (S2) ⎨ ⎩$vₜₜ=\Delta \left(c(t,x)v\right)+{l(t,x)\left|v\right|}^m+k(t,x){\left|u\right|}^q$, t > 0, $x\in {ℝ}^N$, (S3) ⎧$uₜₜ=\Delta \left(a(t,x)u\right)+\Delta \left(b(t,x)v\right)+{h(t,x)\left|u\right|}^p$, t > 0, $x\in {ℝ}^N$, ⎨ ⎩$vₜₜ=\Delta \left(c(t,x)v\right)+{k(t,x)\left|v\right|}^q$, t > 0, $x\in {ℝ}^N$, in $(0,\infty )\times {ℝ}^N$ with u(0,x) = u₀(x), v(0,x) = v₀(x), uₜ(0,x) = u₁(x), vₜ(0,x) = v₁(x). We show that, in each case, there exists a bound B on N such that for 1 ≤ N ≤ B solutions to the systems blow up in finite time.