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On a decay property of weak solution for semilinear evolution equation of parabolic type and its applications. (English) Zbl 0609.35010

Let X be a Hilbert space with the norm \(\| \|\). Suppose that A is a non-negative self-adjoint operator in X. We set \(V=D(A^{1/2})\). Equipped with the norm \(\|| u\|| ^ 2=\| u\| ^ 2+\| A^{1/2}u\| ^ 2\), V is a Hilbert space. For \(a\in X\) and \(f\in L^ 1(0,\infty;X)\), we consider the following initial value problem; \[ (E)\quad du(t)/dt+Au(t)+Nu(t)=f(t),\quad t>0;\quad u(0)=a. \] Here, N is a continuous mapping from V into \(V^ *\) satisfying \[ | <Nu,\phi >| \leq L(\| u\|)\|| u\|| \| A^{1/2}u\| \| A^{1/2}\phi \| \] for all u and \(\phi\) in V, \(<,>\) being the duality between \(V^ *\) and V and \(L=L(\lambda)\) being a monotone increasing function of \(\lambda\) \(\geq 0.\)
In this article, we show that under the assumption \(0\not\in \sigma _ p(A)\) every weak solution u of (E) has the decay property \[ \| (1+A)^{-1/4}u(t)\| \to 0\quad as\quad t\to \infty. \] This is the generalization of the result of K. Masuda [Tôhoku Math. J., II. Ser. 36, 623-646 (1984; Zbl 0568.35077)]. We show also that this result is applicable to the M.H.D. equations and some reaction diffusion equations.

MSC:

35B40 Asymptotic behavior of solutions to PDEs
35K55 Nonlinear parabolic equations
35Q30 Navier-Stokes equations

Citations:

Zbl 0568.35077
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References:

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