Download Blow-up in Nonlinear Sobolev Type Equations (De Gruyter by Alexander B. Al’shin, Maxim O. Korpusov, Alexey G. PDF

By Alexander B. Al’shin, Maxim O. Korpusov, Alexey G. Sveshnikov

The monograph is dedicated to the examine of initial-boundary-value difficulties for multi-dimensional Sobolev-type equations over bounded domain names. The authors reflect on either particular initial-boundary-value difficulties and summary Cauchy difficulties for first-order (in the time variable) differential equations with nonlinear operator coefficients with appreciate to spatial variables. the most objective of the monograph is to procure enough stipulations for international (in time) solvability, to acquire enough stipulations for blow-up of options at finite time, and to derive top and decrease estimates for the blow-up time. The monograph encompasses a giant checklist of references (440 goods) and offers an total view of the modern cutting-edge of the mathematical modeling of assorted very important difficulties coming up in physics. because the checklist of references comprises many papers that have been released formerly in basic terms in Russian learn journals, it could actually additionally function a advisor to the Russian literature.

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Extra resources for Blow-up in Nonlinear Sobolev Type Equations (De Gruyter Series in Nonlinear Analysis and Applications)

Example text

57) @x3 t d exp . t //u. 31). Note that Eq. 57) differs from all equations obtained in this monograph because it takes into account the time dispersion of the dielectric permittivity tensor; the paper [228] is devoted to such equations. We also note that in works of Yu. D. Pletner, the operator method for analyzing linear Sobolev-type equations of arbitrary time order was proposed. 2 Model pseudoparabolic equations equations and certain elliptic equations. He proved that the Sobolev-type equations can be represented as elliptic equations regularly disturbed by Volterra convolution operators.

18) 2 where TO e is the temperature of free electrons and "2 is a small parameter. 2) depends on the density of sources or sinks of free electrons and has the form similar to the distributions of free and bound electrons of the lattice main centers of the semiconductor. 19) where < 0 for donor impurity centers and > 0 for acceptor impurity centers, respectively. Obviously, D 0 holds in the absence of impurity centers. 1 Models of quasi-stationary processes in semiconductors 25 Now we consider possible boundary conditions for Eqs.

Here we assume the local, unique solvability of these problems in a required smoothness class, although, certainly, the proof of the local-on-time existence and the uniqueness of a solution with required smoothness is, as a rule, an exceedingly complicated problem. However, here we demonstrate only one method of the proof of the blow-up of solutions, which was successfully applied in [229, 230], where many theorems on the local-on-time unique solvability were proved. First, we consider the first initial-boundary-value problem for Eq.

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