151 research outputs found

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

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    Consider the equation uβ€²(t)βˆ’Ξ”u+∣u∣ρu=0,u(0)=u0(x),(1), u'(t)-\Delta u+|u|^\rho u=0, \quad u(0)=u_0(x), (1), where uβ€²:=dudt u':=\frac {du}{dt}, ρ=const>0, \rho=const >0, x∈R3x\in \mathbb{R}^3, t>0t>0. Assume that u0u_0 is a smooth and decaying function, βˆ₯u0βˆ₯β€…=sup⁑x∈R3,t∈R+∣u(x,t)∣.\|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)\|0doesnotdependon does not depend on x,t$

    Linear ill-posed problems and dynamical systems

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    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

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    An explicit formula is derived for the electromagnetic (EM) field scattered by one small impedance particle DD of an arbitrary shape. If aa is the characteristic size of the particle, Ξ»\lambda is the wavelength, a<<Ξ»a<<\lambda and ΞΆ\zeta is the boundary impedance of DD, [N,[E,N]]=ΞΆ[N,H][N,[E,N]]=\zeta [N,H] on SS, where SS is the surface of the particle, NN is the unit outer normal to SS, and EE, HH is the EM field, then the scattered field is Esc=[βˆ‡g(x,x1),Q]E_{sc}=[\nabla g(x,x_1), Q]. Here g(x,y)=eik∣xβˆ’y∣4Ο€βˆ£xβˆ’y∣g(x,y)=\frac{e^{ik|x-y|}}{4\pi |x-y|}, kk is the wave number, x1∈Dx_1\in D is an arbitrary point, and Q=βˆ’ΞΆβˆ£S∣iΟ‰ΞΌΟ„βˆ‡Γ—E0Q=-\frac{\zeta |S|}{i\omega \mu}\tau \nabla \times E_0, where E0E_0 is the incident field, ∣S∣|S| is the area of SS, Ο‰\omega is the frequency, ΞΌ\mu is the magnetic permeability of the space exterior to DD, and Ο„\tau is a tensor which is calculated explicitly. The scattered field is O(∣΢∣a2)>>O(a3)O(|\zeta| a^2)>> O(a^3) as aβ†’0a\to 0 when Ξ»\lambda is fixed and ΞΆ\zeta does not depend on aa. Thus, ∣Esc∣|E_{sc}| is much larger than the classical value O(a3)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 aβ†’0a\to 0 and the number M=M(a)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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