151 research outputs found

### Global existence and uniqueness of the solution to a nonlinear parabolic equation

Consider the equation $u'(t)-\Delta u+|u|^\rho u=0, \quad u(0)=u_0(x), (1),$ where $u':=\frac {du}{dt}$, $\rho=const >0,$ $x\in \mathbb{R}^3$, $t>0$.
Assume that $u_0$ is a smooth and decaying function, $\|u_0\|\:=\sup_{x\in
\mathbb{R}^3, t\in \mathbb{R}_+} |u(x,t)|.$ It is proved that problem (1) has
a unique global solution and this solution satisfies the following estimate
$$\|u(x,t)\|0$does not depend on$x,t$

### Linear ill-posed problems and dynamical systems

A linear equation Au=f (1) with a bounded, injective, but not boundedly
invertible linear operator in a Hilbert space H is studied. A new approach to
solving linear ill-posed problems is proposed. The approach consists of solving
a Cauchy problem for a linear equation in H, which is a dynamical system,
proving the existence and uniqueness of its global solution u(t), and
establishing that u(t) tends to a limit y, as t tends to infinity, and this
limit y solves equation (1). The case when f in (1) is given with some error is
also studied

### Scattering of electromagnetic waves by small impedance particles of an arbitrary shape

An explicit formula is derived for the electromagnetic (EM) field scattered
by one small impedance particle $D$ of an arbitrary shape. If $a$ is the
characteristic size of the particle, $\lambda$ is the wavelength, $a<<\lambda$
and $\zeta$ is the boundary impedance of $D$, $[N,[E,N]]=\zeta [N,H]$ on $S$,
where $S$ is the surface of the particle, $N$ is the unit outer normal to $S$,
and $E$, $H$ is the EM field, then the scattered field is $E_{sc}=[\nabla
g(x,x_1), Q]$. Here $g(x,y)=\frac{e^{ik|x-y|}}{4\pi |x-y|}$, $k$ is the wave
number, $x_1\in D$ is an arbitrary point, and $Q=-\frac{\zeta |S|}{i\omega
\mu}\tau \nabla \times E_0$, where $E_0$ is the incident field, $|S|$ is the
area of $S$, $\omega$ is the frequency, $\mu$ is the magnetic permeability of
the space exterior to $D$, and $\tau$ is a tensor which is calculated
explicitly. The scattered field is $O(|\zeta| a^2)>> O(a^3)$ as $a\to 0$ when
$\lambda$ is fixed and $\zeta$ does not depend on $a$. Thus, $|E_{sc}|$ is much
larger than the classical value $O(a^3)$ for the field scattered by a small
particle. It is proved that the effective field in the medium, in which many
small particles are embedded, has a limit as $a\to 0$ and the number $M=M(a)$
of the particles tends to $\infty$ at a suitable rate. Thislimit solves a
linear integral equation. The refraction coefficient of the limiting medium is
calculated analytically. This yields a recipe for creating materials with a
desired refraction coefficient

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