Asymptotic Behavior of the Drift-Diffusion Semiconductor Equations

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This paper is an in-depth dynamical-systems study of the constant mobility case of the authors' previous paper [J. Differential Equations {\bf 123} (1995), no.~2, 523--566]. There, uniform pointwise bounds, not depending on t, were derived for situations including this one; hence, the existence of an absorbing set in L∞ was demonstrated. In my review for that paper, it was mentioned that the hypothesis of constant mobilities is incompatible with mixed boundary conditions and semiconductor saturation, if the transition points lie on a straight edge in two dimensions (MOSFET transistor, for example). The authors make extensive use of the results of R. Temam\ [{\it Infinite-dimensional dynamical systems in mechanics and physics}, Springer, New York, 1988; MR0953967 (89m:58056) ]. Following Temam, the authors define a (compact, connected) attracting subset of the absorbing set. The associated semigroup is shown to be differentiable, provided the Dirichlet problem for the Laplacian possesses W1,4 regularity. The final result deals with the estimation of the Hausdorff dimension of the attractor in the function space ∏H1 (the compactness above is in ∏L2). Due to the theory developed for infinite-dimensional dynamical systems by Temam and others, this estimate is implied by a certain time average qm of the trace of the linearized system, implying exponential decay of an m-dimensional volume element. The authors estimate m from the general theory. In fact, as noted in the preliminary chapter of Temam's book, in the favorable cases, one can obtain the estimate, qm≤−κ1mα+κ2, if one has knowledge of the asymptotic distribution of eigenvalues of the linear operator associated with the above trace. In such cases, one has (κ2/κ1)1/α as an upper bound for the Hausdorff dimension m. The assumed W1,4 regularity excludes transition points on a straight edge in two dimensions, but permits such transitions at corners. Although the authors assume a certain expression for the distribution of the eigenvalues of the mixed problem, it appears that a weak asymptotic relation suffices. Now such an inequality must hold by results of R. Beals [Bull. Amer. Math. Soc. {\bf 72} (1966), 701--705; MR0196257 (33 \#4449) ] (see also the equivalent argument in terms of n-widths by the reviewer [Proc. Amer. Math. Soc. {\bf 33} (1972), 367--372; MR0296583 (45 \#5642) ]). Finally, although it does not appear to be emphasized in the literature, the fractal dimension of the attractor here in L2 is 0, by use of results concerning metric entropy of smooth classes. These results are presented by G. G. Lorentz [{\it Approximation of functions}, Holt, Rinehart and Winston, New York, 1966; MR0213785 (35 \#4642) ]. This is a very nice paper! It begins with Temam's book as blueprint, and details the theory for drift-diffusion semiconductor systems.



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