Problema de transmision nolineal con coeficientes dependientes del tiempo

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Electronic Journal of Differential Equations, Vol. 2007(2007), No. 131, pp. 1–13. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp)

A NONLINEAR TRANSMISSION PROBLEM WITH TIME DEPENDENT COEFFICIENTS
˜ EUGENIO CABANILLAS LAPA, JAIME E. MUNOZ RIVERA

Abstract. In this article, we consider a nonlinear transmission problem forthe wave equation with time dependent coefficients and linear internal damping. We prove the existence of a global solution and its exponential decay. The result is achieved by using the multiplier technique and suitable unique continuation theorem for the wave equation.

1. Introduction In this work, we consider the transmission problem ρ1 utt − buxx + f1 (u) = 0 u(0, t) = v(L, t), u(L0 , t) = v(L0, t),
0

in ]0, L0 [×R+ , in ]L0 , L[×R , t > 0,
+

(1.1) (1.2) (1.3) (1.4) (1.5) (1.6)

ρ2 vtt − (a(x, t)vx )x + αvt + f2 (v) = 0 t > 0,
1

bux (L0 , t) = a(L0 , t)vx (L0 , t), ut (x, 0) = u (x), vt (x, 0) = v (x),
1

u(x, 0) = u (x), v(x, 0) = v (x),
0

x ∈]0, L0 [, x ∈]L0 , L[,

where ρ1 , ρ2 are constants; α, b are positive constants, f, g are nonlinear functions and a(x,t) is a positive function. Controllability and Stability for transmission problem has been studied by many authors (see for example Lions [7], Lagnese [5], Liu and Williams [8], Mu˜oz Rivera and Portillo Oquendo [9], Andrade, Fatori and n Mu˜oz Rivera [1]). n The goal of this work is to study the existence and uniqueness of global solutions of (1.1)-(1.6) and the asymptotic behavior of the energy.All the authors mentioned above established their results with constant coefficients. To the best of our knowledge this is a first publication on transmission problem with time dependent coefficients and the nonlinear terms. In general,the dependence on spatial and time variables causes difficulties,semigroups arguments are not suitable for finding solutions to (1.1)-(1.6); therefore,we make use of aGalerkin’s process. Note that the time-dependent coefficient also appear in the second boundary condition, thus there are some technical difficulties that we need to overcome. To prove the exponential decay, the main difficulty is that the dissipation only
2000 Mathematics Subject Classification. 35B40, 35L70, 45K05. Key words and phrases. Transmission problem; time dependent coefficients; stability. c 2007Texas State University - San Marcos. Submitted May 2, 2007. Published October 9, 2007.
1

2

˜ E. CABANILLAS L., J. E. MUNOZ R.

EJDE-2007/131

works in [L0 , L] and we need estimates over the whole domain [0, L]; we overcome this problem introducing suitable multiplicadors and a compactness/uniqueness argument. 2. Notation and statement of results We denote (w, z) =
I

w(x)z(x)dx,|z|2 =
I

|z(x)|2 dx

where I =]0, L0 [ or ]L0 , L[ for u’s and v’s respectively. Now, we state the general hypotheses. (A1) The functions fi ∈ C 1 (R), i = 1, 2, satisfy fi (s)s ≥ 0 for all s ∈ R and |fi (s)| ≤ c(1 + |s|)ρ−j ,
s (j)

∀s ∈ R, j = 0, 1

for some c > 0 and ρ ≥ 1. We assume that f1 (s) ≥ f2 (s) and set Fi (s) =
0

fi (ξ)dξ .

(A2) We assume that the coefficient asatisfies a ∈ W 1,∞ (0, ∞; C 1 ([L0 , L])) ∩ W 2,∞ (0, ∞; L∞ (L0 , L)) at ∈ L1 (0, ∞; L∞ (L0 , L)) a(x, t) ≥ a0 > 0, We define the Hilbert space V = {(w, z) ∈ H 1 (0, L0 ) × H 1 (L0 , L) : w(0) = z(L) = 0; w(L0 ) = z(L0 )} . Also we define the first-order energy functionals associated to each equation, E1 (t, u) = E2 (t, v) = 1 ρ1 |ut |2 + b|ux |2 + 2 2
L0

∀(x, t) ∈]L0 , L[×]0, ∞[ .

F1 (u)dx
0

L 1 2ρ2 |vt |2 + (a, vx ) + 2 F2 (v)dx 2 L0 E(t) = E1 (t, u, v) = E1 (t, u) + E2 (t, v).

We conclude this section with the following lemma which will play essential role when establishing the asymptotic behavior of solutions. Lemma 2.1 ([2, Lemma 9.1]). Let E : R+ → R+ be a non-increasing function and 0 0 assume that there exist two constants p > 0 and c > 0 such that
+∞

E (p+1)/2 (t)dt ≤...
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