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Abstract
This paper concerns the null controllability of a semilinear control system governed by degenerate parabolic equation with a gradient term, where the nonlinearity of the problem is involved with the first derivative. We first establish the well-posedness and prove the approximate null controllability of the linearized system, then we can get the approximate null controllability of the semilinear control system by a fixed point argument. Finally, the semilinear control system with a gradient term is shown to be null controllable.
Highlights
1 Introduction In this paper, we investigate the null controllability of the following semilinear degenerate system: ut – xαux x + g(x, t, u, ux) = h(x, t)χω, (x, t) ∈ QT, (1.1)
We assume that g satisfies g(·, ·, 0, 0) = 0 and gs(x, t, s, p) + x–α/2 gp(x, t, s, p) ≤ K, ∀(x, t, s, p) ∈ QT × R × R, (1.5)
(2020) 2020:55 example of a Crocco-type equation coming from the study on the velocity field of a laminar flow on a flat plate
Summary
For semilinear problem (1.1)–(1.4), the authors showed the regional and persistent regional null controllability in [3, 5]. In [1, 19], the authors proved the null controllability of problem (1.1)–(1.4) with g(x, t, u, ux) = f (x, t, u) and g(x, t, u, ux) = xα/2b(x, t)ux + c(x, t)u, (1.14) We prove the approximate null controllability of linear problem (1.1)–(1.4) with (1.14).
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