Global existence of generalized solution to semi linear wave equation with nonlinear second order boundary damping.

Introduction

Wave propagation is encountered in various real life phenomena particularly in this area of Information and Communication Technology.

Computer experts tried to improve on the multiple accessibility of wireless link with less path loss at the cell edge. Recent researches and considerations had shifted from Code Division Multiplexing Access [CDMA] to Orthogonal Frequency Division Multiplexing [OFDM], [1] [2] [3] [4] in order to obtain maximum transmitting power and to improve the quality of data transmission link.

This can be attained in at most near-barrier free medium. It becomes necessary for Mathematicians to intensify researches on wave problems formulated in a medium with common damping effects which tends to hinder and restrict smooth wave propagations.

In [5] [6] [7], global existence of solution to hyperbolic equation with damping effects had been given either in the equation formulated or at its initial or boundary condition considered.

In [8], we gave the necessary conditions for solvability of a semilinear hyperbolic equations with nonlinear second order boundary damping.

In this paper, we established a global existence of generalized solution to semilinear hyperbolic equation with nonlinear second order boundary damping in the form of an initial boundary value problem thus:

[u.sub.tt] - [summation over (i,j)] [a.sup.i,j] (x,t)[u.sub.xi][u.sub.xj] = f (x,t,u) in [OMEGA]x (0,[infinity]) 1.1

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] 1.2

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] 1.3

u (x,o) = [phi] (x), [u.sub.t] (x,o) = [psi] (x) 1.4

where u[member of][C.sup.2] is the unknown function, the term k(u)[u.sub.tt] models internal forces when the density of the medium depends on the displacement.

Let [OMEGA] be a bounded open set of [R.sup.n] possessing sufficiently smooth boundary [partial derivative][OMEGA] with partition [Y.sub.0], [Y.sub.1] such that [[bar.[gamma]].sub.0] [intersection] [[bar.[gamma]].sub.1] = [phi] and 'n' represents the outward unit normal vector on [gamma].

We shall denote

(i) [H.sub.1] ([OMEGA]) = {u [member of] [H.sup.1] ([OMEGA]); u = 0 on [[gamma].sub.0]} 1.5

(ii) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] 1.6

(iii) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] 1.7

which is a time dependent bilinear form representing the linear combination of various orders of derivatives.

Assumptions and lemma

The following assumptions and lemma were made:

(a) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] 1.8

(b) [absolute value of [a.sub.t](u,u;t)] [less than or equal to] [c.sub.0][[parallel]u[parallel].sup.2][L.sup.p] 1.9

from the uniform symmetric hyperbolicity conditions for all u [member of] [H.sup.1.sub.0] ([OMEGA]).

Let k(u) be a continuous real function, we recalled that for all u[member of] [H.sup.1.sub.0] ([OMEGA]) and 1[less than or equal to]p<n, there exists a constant 'c' depending only on p and n such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] 1.10

where [p.sup.*] is the dual of p.

And following Gagliardo--Nirenberg--Sobolev inequality [9], we have that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] 1.11

thus

(c) (i) 0 [less than or equal to] [k.sub.0] [less than or equal to] k(u) [less than or equal to] [k.sub.1] (l + [[absolute value of u].sup.p])

(ii) [[absolute value of [k.sup.1] (u)].sup.p/p-1] [less than or equal to] [k.sub.2] (l + k(u)) 1.12

[k.sub.0]; [k.sub.1]; [k.sub.2] being some positive real constants [10], [11]

Lemma 1: Let f(x,t,u) be Lipschitz continuous in u with a bounded differential coefficient in u in a relative compact region [OMEGA], then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Proof: Since f is Lipschitzian in u, it implies that for any two arbitrary points (x,t,[u.sub.1]) and (x,t,[u.sub.2]) in [OMEGA], there exists a positive integer L such that

[absolute value of f (x, t, [u.sub.1]) - f (x, t, [u.sub.2])] [less than or equal to] L [absolute value of [u.sub.1] [u.sub.2]]

so for a given [epsilon]>0 and [delta]>0 where [absolute value of [u.sub.1] - [u.sub.2]] < [delta], we have

[absolute value of f (x,t,[u.sub.1]) - f (x, t,[u.sub.2])] < [epsilon]

Thus f is continuous and bounded. By Sobolev's inequality, f [member of] [L.sup.2] (0, T; [L.sup.2] ([OMEGA])) implies that Df [member of] [L.sup.2] (0, T,[L.sup.2] ([OMEGA])) which shows that '[f.sub.u]' exists and is bounded. Let [absolute value of [f.sub.u] u'] be majorized by [absolute value of [f.sub.u]] [absolute value of u']

Then [absolute value of [f.sub.u]u'] [less than or equal to] [absolute value of [f.sub.u]] [absolute value of u']

[??] [absolute value of [f.sub.u]u'] [less than or equal to] [c.sub.2] [parallel]u'[parallel]

for [absolute value of [f.sub.u]] [less than or equal to] [c.sub.2]

Existence of Generalized Solution

The necessary condition in establishing the existence of a generalized solution to the given problem 1.1-1.4 is to obtain a smooth function r [member of] [c.sup.[infinity]] ([OMEGA]) that a tempered distribution can be multiplied without going outside that class. Such function must be a multiplier of the space and the multiplication will be a continuous linear operator equivalent to the estimates.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.1)

Where [alpha] is arbitrary multi-index and [C.sub.[alpha]], [N.sub.[alpha]] are constants [12]

Definition 1.1 A tempered distribution are those set of functions (elements) of the dual space to the Frechet space S ([R.sup.N]) where notion of compact supports are replaced by rapid decay at infinity and such functions satisfy the inequality

[absolute value of f (x)] [less than or equal to] [c.sub.N] [(1 + [absolute value of x]).sup.N]

Like the ordinary measurable functions.

Further abstract proof of existence is based on Galerkin's method.

Finite Dimensional Space Formulation and Approximate Solution

We Consider the Space X = [H.sub.1] ([OMEGA]) [intersection] [L.sup.p+2] ([gamma]) which is separable.

Let {[w.sub.j]}j [greater than or equal to] 1 be a basis and [w.sub.j] (x), J = 1,2 ... being smooth multipliers such that the sequence {[w.sub.j](x)}j [greater than or equal to] 1 is an orthogonal basis of [H.sup.1.sub.0] ([OMEGA]) and orthonormal basis of [L.sup.2] ([OMEGA]).

We assume that there exists a unique function '[u.sub.m]' in X defined by

[u.sub.m] = [m.summation over (j=1)] [g.sub.jm] (t)[[omega].sub.j] (2.2)

Satisfying of the initial conditions

[g.sub.jm] (0)=([phi],[[omega].sub.j])=[m.summation over (j=1)] ([phi], [[omega].sub.j]) [[omega].sub.j] (2.3)

[g'.sub.jm] (0)=([psi],[w.sub.j])=[m.summation over (j=1)] ([psi], [w.sub.j])[w.sub.j] (2.4)

each converging strongly in [H.sup.1.sub.0] ([OMEGA]) [intersection] [H.sup.2] as m [right arrow] [infinity] to [phi] and [psi] respectively.

The value [g.sub.jm] (0) is the jth component of [u.sub.m] while [u.sub.om] is the orthogonal projection of [phi] and similarly [u'.sub.om] being of the orthogonal projection of [psi] on the space spanned by [[omega].sub.j] ; x [greater than or equal to] 1.

In [8], we prove that '[u.sub.m]' actually exists and is unique also the coefficients [g.sub.jm] (t),0 [less than or equal to] t [less than or equal to] [t.sub.m] < T were selected to satisfy the equation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.5)

which resulted in

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.6)

The solution [U.sub.m] to (2.6) becomes the approximate solution in the finite dimensional subspace while [[omega].sub.j] (x) are the normalized eigen function.

The corresponding formulation of equation 1.1-1.4 in finite dimensional space has the form

[u".sub.m] - [m.summation over (i,j=1)] ([a.sup.i,j] (X, t)[u.sub.m], [x.sub.i])[x.sub.J] = [p.sub.m] (f (x, t, [u.sub.m])) (2.7)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.8)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.9)

[u.sub.m] (x, o) = [p.sub.m] [phi](x), [u'.sub.m] (x,0) = [p.sub.m] [psi](x) (2.10)

where [p.sub.m] is the orthogonal projector or injector (a linear or nonlinear operator) unto the space. The matrix of p depends on the problem arising from various independency of the unknown. It is important to note that ([u".sub.m],[[omega].sub.j])=[g".sub.m](t) and [summation][g".sub.m](t)[[omega].sub.j] = [u".sub.m] we now extend these approximate solution to the entire space and pass to the limit as m [right arrow] [infinity] by applying Gronwall's inequality and a priori estimates to the solution representation.

A Priori estimates

We multiply equation 2.6 by g "m (t) and sum over j to obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.11)

each of the terms are estimated thus

1. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.12a)

2. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.12b)

3. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.12c)

4. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.12d)

with l > 0 arising from Cauchy--Schwarz inequality. Applying these inequalities

(a-d) in equation 2.11 with the assumptions, we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.13)

where we denote by L all constants independent of m and fixed [epsilon]=1/2 in order to apply the Gronwall's inequality. We define

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

then equation 2.13 becomes

[E'.sub.m] (t) [less than or equal to] [l.sub.1] [E.sub.m] (t) [E.sub.m] (t) + [l.sub.2] [[parallel]f[parallel].sup.2] 0 [less than or equal to] t [less than or equal to] T (2.14)

with certain additive terms of negligible bounds.

Thus

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.15)

and (2.15) becomes

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Next, we differentiate 2.6 with respect to 't' and multiply with [g".sub.jm] (() summing over j, j [greater than or equal to] 1,m to obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.16)

The terms of equation (2.16) can also be estimated as in equation (2.11) thus

5. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

6. k ([u.sub.m]) [u".sub.m] [u"'.sub.m] = [d/dt] ([1/2] k ([u.sub.m]) [([u".sub.m]).sup.2]) - [1/2] k' ([u.sub.m])[u'.sub.m] [([u".sub.m]).sup.2]

such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

7. [absolute value of ([f.sub.u][u'.sub.m],[u".sub.m])] [less than or equal to] [absolute value of c ([u'.sub.m],[u".sub.m])] [less than or equal to] [d/dt] ([1/2] c[[absolute value of [a'.sub.m]].sup.2])

substituting these estimates into equation (2.16) we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.17)

we define

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

then equation (2.17) reduces to

[E.sub.2m](t) [less than or equal to] [m.sub.1] + [m.sub.2] [[parallel]f[parallel].sup.2] 0 [less than or equal to] t [less than or equal to] T (2.18)

with some additive constants; [m.sub.11] [m.sub.2] are constants. By applying the Gronwall's inequality to equation (2.18) resulted to

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.19)

since 0 [less than or equal to] t [less than or equal to] T is arbitrary, we conclude that both [E.sub.m](t) and [E.sub.2m](t) reduces to a constants such that all the additive constants not specifically covered by the estimates are bounded.

Thus, this concludes the boundedness of the weak solution '[u.sub.m]' to equation 1.1-1.4 in a finite dimensional space.

Passage to the Limit

We now consider an arbitrary function [eta] [member of] C'((0, T), [H.sup.1.sub.0] ([OMEGA])) such that [eta](T) = [[eta].sup.1] (T) = 0 of the form

[eta] (t) = [m.summation over (k=1)] [g.sup.k] (t)[w.sub.k]

where [g.sup.k] are smooth multipliers functions of the space.

we multiply equation 2.6 by [eta] (t) and integrate by parts over the interval (0, T) to get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.20)

implying

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.21)

Since the space is of compact support and taking into consideration that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], the estimates and boundedness of the solution representation showed that

[u.sub.m] is bounded in [L.sup.[infinity]] (0,T;H'([OMEGA])),

[u'.sub.m] is bounded in [L.sup.[infinity]] (0, T; H'([OMEGA]) [intersection] [L.sup.p+2] ([gamma])),

[u".sub.m] is bounded in [L.sup.[infinity]] (0, T; H'([OMEGA]) [intersection] [L.sup.p+2] ([[gamma].sub.1]))

hence, there exist subsequences [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] from [u.sub.m] where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Following the Hahn-Banach theorem to extend the map [p.sub.m] [member of] L([y.sub.k], [y.sub.k])to the entire space L(y,y), we have that the subsequences together with their corresponding derivatives converges in the following way

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.22)

Definition 1.2: If 1 [less than or equal to] p < n, the Sobolev Conjugate of p is [p.sup.*] = [np/n - p] for

[1/p*] - [1/n], p* > p

Such that for p = 1, p* = q and [1/p] + [1/q] = 1.

Thus from Gagliardo--Nirenberg--Sobolev inequality [9] where r is chozen such that [rn/n - 1] = (r - 1) [ p/p - 1]

Then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] holds as in equation 1.10.

we observed that for the space [L.sup.p+2] ([OMEGA]) the conjugate is such that q = [p + 1/p + 2] and

[absolute value of [u'.sub.m]] [u'.sub.m], k ([k.sub.m]) [u'.sub.m], k' ([k.sub.m]) [u'.sub.m] [member of] [L.sup.q] ([gamma])

Therefore [u.sub.m] [right arrow] u a.e on [gamma]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.23)

Since [[omega].sub.j] is dense in X = [H.sub.1] ([OMEGA]) [intersection] [L.sup.p+2]([gamma]), there exists an arbitrary smooth function v [member of] [H.sub.1] ([OMEGA]) [intersection] [L.sup.p+2] ([gamma]) such that the relation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.24)

Global Existence of Generalized Solution to Semi Linear Wave Equation 9

Main Result

Considering a semi-linear hyperbolic problem of the form 1.1-1.4 in an infinite dimensional space, this paper guarantee the existence of solution and we state as theorem.

Theorem: (Existence of a generalized solution)

Let the coefficient of a semi linear IBVP of the form 1.1-1.4 satisfy the assumptions a,b,c. If there exists a finite dimensional formulation of 1.1 with bounded, convergent approximate solutions and for arbitrary smooth functions v [member of] [H.sub.1] ([OMEGA]) [intersection] [L.sup.p+2] ([gamma]), [eta](t) [member of] C'[0, T] with [eta](T) = 0 then there exist a generalized solution satisfying the integral identity (2.24) provided Lemma 1 holds.

References

[1] Zaher, D, Sasa, D & Ionnis O: (2004): Coverage and capacity Enhancement of CDMA cellular systems via Multihop Transmission. Institute of Comm. Engineering (LNT) Germany.

[2] J. Y. Baudais and M. Crussiere (2007): Resource Allocation with adaptive spread spectrum OFDM using 2D spreading of powerlilne communications. EURASIP Journal on Advances in signal processing.

[3] S. Antoine, G. Emeric, C. Mathier, Y. B. Jean and F. H. Jean (2008): Optimization of Linear Precoded OFDM for High-Data-Rate UWB Systems. EURASIP Journal of Wireless Communication.

[4] Joseph Isabona and Moses Ekpenyong (2009): Performance Evaluation of coverage-capacity interaction in CDMA Wireless Networks Journal of the Nig. Mathematical Society Vol. 28, p 153=167.

[5] M. Aassila (2001): Global Existence of solutions to a wave equation with damping and source terms. Differential and integral Equations 14: 1302-1314.

[6] M. Nakao and K. Ono (1993): Existence of Global Solution to the Cauchy problem for semi-linear dissipative wave equations. Maths. Z. 214: 325-342.

[7] M. Nakao and K. Ono (1995), Global Existence to the Cauchy problem of the semi-linear wave equation with a non-linear dissipative. Funkcialaj Ekvacioj 38; 417-431

[8] Ndiyo, Etop E (2009): Necessary Conditions for Solvability of hyperbolic Equations in infinite Dimensional Space. World Journal of Applied Science and Technology vol.1 no.1:122-126

[9] Evans, L. C (1998): Differential Equations--Graduate studies in Mathematics vol. 19 American Maths. Society, Providence Rhole, Island.

[10] Doronin G. G, Lar'kin, N.A Sousa, A. J. (1998): A hyperbolic problem with Nonlinear second order Boundary Damping. Electronic Journal of Differential Equation Vol. 1998 No. 28: 1-10.

[11] Jong Yeoul Park and Sun Hye Park (2003): Solution for a hyperbolic System with boundary differential inclusion and nonlinear second order boundary damping. Electronic Journal of Differential Equations Vol. 2003 No. 80: 1-7

[12] Wloka J. (1987). Partial Differential Equation Cambridge University Press.

Etop E. Ndiyo

Department of Mathematics University of Uyo, Uyo

E-mail: ndiyo2008@yahoo.com