Language：English   Publishing type：Research paper (scientific journal)   Publisher：SPRINGER BASEL AG  

In this paper, the initial value problem for the drift-diffusion equation which stands for a model of a semiconductor device is studied. When the dissipative effect on the drift-diffusion equation is given by the half Laplacian, the dissipation balances to the extra force term. This case is called critical. The goal of this paper is to derive decay and asymptotic expansion of the solution to the drift-diffusion equation as time variable tends to infinity.

DOI： 10.1007/s00023-015-0428-7

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：ACADEMIC PRESS INC ELSEVIER SCIENCE  

The initial-value problem for the dissipative equation with the extra-force-term is considered. The dissipative effect on this equation is given by the fractional Laplacian which describes the anomalous diffusion. The main goal of this paper is to give an estimate on the difference between solutions and their asymptotic expansion as the space and the time variables tend to infinity. Furthermore, as an application, the large-time behavior of solutions to some concrete problems are discussed. (C) 2015 Elsevier Inc. All rights reserved.

DOI： 10.1016/j.jmaa.2015.02.025

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：ACADEMIC PRESS INC ELSEVIER SCIENCE  

We study local and global existence and uniqueness of solutions to the drift-diffusion equation with fractional dissipation (-Delta)(theta/2). In the preceding works for some associated equations, the cases theta = 1 and theta < 1 are known as critical and supercritical respectively. In the critical and supercritical cases, we may not apply the L-p-theory for semilinear equations of parabolic type used in the subcritical case 1 < theta <= 2. We discuss local existence with large data and global existence with small data in the Besov space B-p,q(n/p-theta) (R-n), which corresponds to the scaling invariant space of the equation. Furthermore we show that solutions can blow up in finite time if initial data is not small. (C) 2015 Elsevier Inc. All rights reserved.

DOI： 10.1016/j.jde.2014.12.033

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：HIROSHIMA UNIV, GRAD SCH SCI  

The initial value problem for the drift-diffusion equation arising from a model of semiconductor-devices is studied. The goal in this paper is to derive wellposedness and real analyticity of solutions of the initial value problem for the drift-diffusion equation with its dissipating term A = (-Delta)(1/2). In the preceding works for some associated equations, the case corresponding to this is known as critical. In this case, the drift-diffusion equation A with L is of elliptic type, so we may not apply the L-p-theory for parabolic partial differential equations used in the case that the dissipating term is A(theta) = (-Delta)(theta/2) with 1 < theta <= 2.

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：EDP SCIENCES S A  

We study the initial value problem for the drift-diffusion model arising in semiconductor device simulation and plasma physics. We show that the corresponding stationary problem in the whole space R-n admits a unique stationary solution in a general situation. Moreover, it is proved that when n >= 3, a unique solution to the initial value problem exists globally in time and converges to the corresponding stationary solution as time tends to infinity, provided that the amplitude of the stationary solution and the initial perturbation are suitably small. Also, we show the sharp decay estimate for the perturbation. The stability proof is based on the time weighted L-p energy method.

DOI： 10.1051/cocv/2011191

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：KHAYYAM PUBL CO INC  

We consider the Nernst-Planck-type drift-diffusion equation with fractional dissipation. For the initial-value problem of this equation, the well-posedness, the time-global existence, and the decay of solutions were already shown. When the dissipation operator is given by the Laplacian, the asymptotic expansion of the solution as t -> infinity was obtained in a previous paper. We also derive the asymptotic expansion of the solution to the drift-diffusion equation with the fractional Laplacian.

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：SIAM PUBLICATIONS  

In this paper, the Cauchy problem for the linear dissipative equation with a potential is studied. The dissipative effect of this equation is given by the fractional Laplacian. When a dissipative equation is considered, the fractional Laplacian describes the anomalous diffusion. The main goal of this paper is to derive the large-time behavior of decaying solutions. Particularly, the estimate on the difference between solutions and their asymptotic expansion as t -> infinity is given. The spatial decay of this difference is also derived. Generally speaking, when a dissipative equation with the fractional Laplacian is studied, it is difficult to obtain the asymptotic expansion of solutions with high order. The anomalous diffusion causes this difficulty. The spatial decay of the difference between solutions and their asymptotic expansion provides the asymptotic expansion with arbitrary high order. Furthermore, as an application, the large-time behavior of solutions to a nonlinear problem is discussed.

DOI： 10.1137/120873200

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：BIRKHAUSER VERLAG AG  

We consider the Cauchy problem for the drift-diffusion equation arising from the model of semiconductor devices. We know the well-posedness, the time global existence and the decay of solutions. We show the spatial analyticity of the solution.

DOI： 10.1007/s00013-011-0302-x

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：ACADEMIC PRESS INC ELSEVIER SCIENCE  

We consider the large-time behavior of the solution to the initial value problem for the Nernst-Planck type drift-diffusion equation in whole spaces. In the L(p)-framework, the global existence and the decay of the solution were shown. Moreover, the second-order asymptotic expansion of the solution as t -> infinity was derived. We also deduce the higher-order asymptotic expansion of the solution. Especially, we discuss the contrast between the odd-dimensional case and the even-dimensional case. (C) 2010 Elsevier Inc. All rights reserved.

DOI： 10.1016/j.jmaa.2010.02.049

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：Kyoto University  

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：WORLD SCIENTIFIC PUBL CO PTE LTD  

We show the global existence and asymptotic behavior of solutions for the Cauchy problem of a nonlinear parabolic and elliptic system arising from semiconductor model. Our system has generalized dissipation given by a fractional order of the Laplacian. It is shown that the time global existence and decay of the solutions to the equation with large initial data. We also show the asymptotic expansion of the solution up to the second terms as t ->infinity.

DOI： 10.1142/S021820250900367X

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Event date： 2019.6

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