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 $u_0 \in L_{1, loc} ({\mathbb R}^n)$

QUALITY OF LIFE IN TYPE-I DIABETES MELLITUS PATIENTS WITH DIABETIC VEGETOPATHY

"Quality of life" represents one of the most important parameters of individual coping the problems of social adaptation in case of persisting manifestations of the basic disease. A total of 46 type I diabetes mellitus patients with diabetic vegetopathy were examined. Diagnosis was anamnestically and instrumentally verified. Patients' quality of life was investigated using MMP1 in a Kincannon's variant, Zungs depression scale, and Spitcer's quality-of-life scale. The following code was established: schizoidia - psychasthenia - hypochondria - paranoia. According to the depression scale, there was a slight depression in 51,7 per cent of the patients, a moderate - in 36,5 per cent but a severe - in 13,7 per cent. According to the quality-of-life scale, the parameter of self-confidence was characterized as poor in 17,3 per cent of the cases, moderately good - in 31 per cent but good - in 51,7 per cent. Social adaptation was poor in 6,8 per cent of the patients, moderately good - in 31 per cent but good - in 62,2 per cent. The correlation between the expression of vegetopathy, the severity of diabetes mellitus, the self-confidence, social adaptation and "social inertia" was discussed