Localized peaking regimes for quasilinear parabolic equations

This paper deals with the asymptotic behavior as $t\rightarrow T<\infty$ of all weak (energy) solutions of a class of equations with the following model representative: \begin{equation*} (|u|^{p-1}u)_t-\Delta_p(u)+b(t,x)|u|^{\lambda-1}u=0 \quad (t,x)\in(0,T)\times\Omega,\,\Omega\in{R}^n,\,n>1, \end{equation*} with prescribed global energy function \begin{equation*} E(t):=\int_{\Omega}|u(t,x)|^{p+1}dx+ \int_0^t\int_{\Omega}|\nabla_xu(\tau,x)|^{p+1}dxd\tau \rightarrow\infty\ \text{ as }t\rightarrow T. \end{equation*} Here $\Delta_p(u)=\sum_{i=1}^n\left(|\nabla_xu|^{p-1}u_{x_i}\right)_{x_i}$, $p>0$, $\lambda>p$, $\Omega$ is a bounded smooth domain, $b(t,x)\geq0$. Particularly, in the case \begin{equation*} E(t)\leq F_\mu(t)=\exp\left(\omega(T-t)^{-\frac1{p+\mu}}\right)\quad\forall\,t0,\,\omega>0, \end{equation*} it is proved that solution $u$ remains uniformly bounded as $t\rightarrow T$ in an arbitrary subdomain $\Omega_0\subset\Omega:\overline{\Omega}_0\subset\Omega$ and the sharp upper estimate of $u(t,x)$ when $t\rightarrow T$ has been obtained depending on $\mu>0$ and $s=dist(x,\partial\Omega)$. In the case $b(t,x)>0$ $\forall\,(t,x)\in(0,T)\times\Omega$ sharp sufficient conditions on degeneration of $b(t,x)$ near $t=T$ that guarantee mentioned above boundedness for arbitrary (even large) solution have been found and the sharp upper estimate of a final profile of solution when $t\rightarrow T$ has been obtained.Comment: 27 page

On blow-up conditions for nonlinear higher order evolution inequalities

We obtain exact conditions for global weak solutions of the problem \left\{ \begin{aligned} & u_t - \sum_{|\alpha| = m} \partial^\alpha a_\alpha (x, t, u) \ge f (|u|) \quad \mbox{in } {\mathbb R}_+^{n+1} = {\mathbb R}^n \times (0, \infty), & u (x, 0) = u_0 (x) \ge 0, \end{aligned} \right. to be identically zero, where $m$ and $n$ are positive integers, $a_\alpha$ and $f$ are some functions, and $u_0 \in L_{1, loc} ({\mathbb R}^n)$

On global solutions of quasilinear second-order elliptic inequalities

We consider the inequality - \operatorname{div} A (x, \nabla u) \ge f (u) \quad \mbox{in } {\mathbb R}^n, where $n \ge 2$ and $A$ is a Caratheodory function such that C_1 |\xi|^p \le \xi A (x, \xi) \quad \mbox{and} \quad |A (x, \xi)| \le C_2 |\xi|^{p-1} with some constants $C_1 > 0$, $C_2 > 0$, and $p > 1$ for almost all $x \in {\mathbb R}^n$ and for all $\xi \in {\mathbb R}^n$. Our aim is to find exact conditions on the function $f$ guaranteeing that any non-negative solution of this inequality is identically zero

Collapses and revivals of polarization and radiation intensity induced by strong exciton-vibron coupling

Recently, systems with strong coupling between electronic and vibrational degrees of freedom attract a great attention. In this work, we consider the transient dynamics of the system consisting of strongly coupled vibron and exciton driven by external monochromatic field. We show that under coherent pumping, polarization of exciton exhibits complex quantum dynamics which can be divided into three stages. At the first stage, exciton oscillations at its eigenfrequency relax due to the transition to set of shifted Fock states of vibrons. We demonstrate that these shifted Fock states play the role of an effective reservoir for the excited exciton state. The time of relaxation to this reservoir depends on exciton-vibron coupling. At the second stage, excitation, transferred to the reservoir of the vibronic shifted states at the first stage, returns into electronic degrees of freedom and revival of oscillations at exciton eigenfrequency appears. Thus, the dynamics of molecular polarization exhibit collapses and revivals. At the final stage, these collapses and revivals dissipate and polarization exhibits Rayleigh response at the frequency of the external field. Discovered collapses and revivals manifest in radiation spectrum as multiple splitting of the spectral line near the exciton transition frequency