## On uniqueness and stability of steady-state carrier distributions in semiconductors.(English)Zbl 0609.35024

Differential equations and their applications, Equadiff 6, Proc. 6th Int. Conf., Brno/Czech. 1985, Lect. Notes Math. 1192, 209-214 (1986).
[For the entire collection see Zbl 0595.00009.]
The author considers a system of nonlinear partial differential equations in the form $(1)\quad \Delta u=-(q/\epsilon)[f+p-n],\quad R=(np-n^ 2_ i)/[\tau (n+p+2ni)]$
$(2)\quad qn_ t=\nabla \cdot J_ n- qR,\quad J_ n=q\mu_ n(k\nabla n-n\nabla u)$
$(3)\quad qp_ t=- \nabla \cdot J_ p-qR,\quad J_ p=-q\mu_ p(k\nabla p+p\nabla u)$ subject to the following boundary and initial conditions:
(4a) $$u=U_ s$$, $$n=N_ s$$, $$p=P_ s$$ on $$S_ 1$$ for $$t>0$$
(4b) $$\nu \cdot \nabla u=\nu \cdot \nabla n=\nu \cdot \nabla p=0$$ on $$S_ 2$$ for $$t>0$$
(5) $$n(0,x)=n_ 0(x)$$, $$p(0,x)=p_ 0(x).$$
Here u(t,x), n(t,x) and p(t,x) are to be solved while the functions f(t,x) $$U_ s(t,x)$$, $$N_ s(t,x)$$, $$P_ s(t,x)$$, $$n_ 0(x)$$, $$p_ 0(x)$$ and the constants q, $$\epsilon,n_ i,\tau,\mu_ n,\mu_ p$$ are known. The point x is supposed to be in a bounded Lipschitzian domain G in $$R^ d$$ (d$$\leq 3)$$, whose boundary S consists of the union of the disjoint parts $$S_ 1$$ and $$S_ 2$$. $$\nu$$ in (4b) denotes the unit normal vector to S. This problem arises in the theory of the carrier distributions in semiconductors. One of the results obtained in the paper (Theorem 1) shows that under certain conditions the stationary solution $$u=U(x)$$, $$p=P(x)$$, $$n=N(x)$$ of the problem (1-4) is unique in the space $$H^ 1_ 2\cap L_{\infty}$$ if $$p\geq 0$$ and $$n\geq 0$$. The second result claims that under certain conditions the time-dependent solution u(t,x), p(t,x) and n(t,x) of the problem (1-5) converges to the above mentioned stationary solution exponentially in time.
Reviewer: M.Idemen

### MSC:

 35G30 Boundary value problems for nonlinear higher-order PDEs 35B40 Asymptotic behavior of solutions to PDEs 35B35 Stability in context of PDEs 35G25 Initial value problems for nonlinear higher-order PDEs

Zbl 0595.00009
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