Degenerated nonlinear hyperbolic equation with discontinuous coefficients

Transcription

1 Pro~ Indian Acad. Sci. (Math. SO_), VoL 94, Nos 2 & 3, December 1985, pp Printed in India. Degenerated nonlinear hyperbolic equation with discontinuous coefficients M E KHALIFA 12, Mob9 Shalaby Street, H-,J~ k EI-Kuha, Egypt MS received 11 Jfily 1985 Almmet. The ~ diff~-e~.~d equations with discontinuous coel~ients have been extended in various dimctiom by 9 number of authors [2], [3], [11], [13], [14], [16"J. This paper deals with a mixed problem for a dzge~mted nonlinear hyperbolic equation with discontinuous aceffg~ntx geywm'd~ Partial differential equgfion; discontinuous r eqmtion. nonlinear hyperbolic 1. Statement of the problem and mtittioas Let Q = D x [0, T] where D is a bounded domain in R" with a smooth boundary F, F~ separates Dinto two domains D1 and D2,y = Fl [0, T],S --- F[0, T'J, QI = DI[0, T'J and Qz --- D2 [0, T]. Consider the hyperbofic equation with initial conditions f(t)[u n- ~ (at~(x)ux,)x,] + ~ b,(x, t)ux, = F(x, t, u) (1) i.j~l i~l u(x,o) = u,(x, 0) = 0, (2) boundary condition u(x, OIs -- 0, (3) and jumping conditions u(x, t)l~+o = u(x, t)l,-o, (4) where x = (xl... x.), k~ is a positive constant, K(x) = ~ Kl(x), L K, (x), x~dt x E Dz' 129

2 130 M E Khalifa d,~, d - - = 2., ao c~ (v, xl) dn ~.~ = ~ r and v is the exterior unit normal of D~. For studying the problem (1)--(4) we have to define the following spaces; W~ (Q) = W~ (QI)+ W~ (Qt) is a Sobolev space (see [I]);, E(Q) is a set of functions defined in Q and satisfy o It, x) f(t) < oo. ff q We will refrain from writing the symbols dx and dxdt. DEFINmON. The function u(x, t) is called the solution of (1)-(4), if it satisfies tkix)u.*+ f; q l.j=l 1 " btu~xcb_~(t) I,j=l K(X)CbF(x,t,u)] and (2), (3) in L 2. (5) 2. Assumption and results THEOREM. Suppose that (i) f(t), t > 0 is continuous, differentiable and for small t satisfies Mt t* <~ f(t) <~ M2t~; M~t "-t <~ f'(t) <~ M4t ~-l, where 0 ~< ~ < 1 M~, l = 1, 2... are positive constants. (ii) Zr.j= t a0 ~t~ >/2Z~'= t ~,). >0 and Obi/at is continuous in Q1, Q2- (iii) F(x, t, u) is measurable with respect to x, continuous with respect to t and has bounded derivatives F, F., in QI,Q2 and F(x, t, u) <~ p(x, t)u +q(x, t) where p, q = 0(t" + 3/2), Ft = 0(t ~ + ~/2), then there exists a unique weak solution to problem (I)-(4). Proof. Let v,(x) be a linear independent system in W~ (D), satisfy condition (4) and compact in W, l (D). We seek the solution of problem (1)-(4) in the form (see 1"9]): u.(x,t) = ~ c,(t)v,(x). r=l

3 Hyperbolic equation with discontinuous coefficients 131 Substitute u(x, t) with u.. (x, t) in (1), multiply by v,(x) and integrate over D; the result is the following system of ordinary differential equations: f (t) r~= l Cr (t) f D KOr Vs + f (t) r~l Cr (t) ~D i,j~= l aij(~r)x~ (P$)x~ K + f(l) r~= l Cr(t) ~F k3vru. +r~= l Cr(t) fd i~= n bi(~ + f(t),~ c,(t)" au(v,)x, Kx, = i,j=l =IoKF(x,t ~ c,v,)v., (6) s = (1... m) with initial conditions, c,(o) = c,(o)= o. (7) The existence of the solution to the problem (6)--(7) was proved (see [6], [12]). Now, we will prove that u., (x, t) converges to u(x, t) which is the weak solution to problem (1)-(4). To do so, we will lind some important a priori estimates. This is usually where the hard work comes in. Many interesting estimates are discussed in [5]-[8] and [ 15]. Let r (x, t) be an infinitely differentiable function which tends to zero in Q - Q,, r = I in Q2, and Irhl ~< 1 in Q2, - Q,, where e > 0. Substitute F with F,, where F, = rhf in system (6) to get the following system: f(t)~lc/;okv, v,+f(t)~=tc, fo,j=~, ao(v')x,(o')x~ K + f(t) ~ c, ~1- k3v'v'+ f(t), ~ c, fo ~ a~ (K)x'V" r=l I =1 ij=i (8) For convenience, we will adopt the notation ofu, to be understood as u,.,. Multiply (8) by c'., l/f(t)t p + 2~, 0 < fl < 1 - a; sum over s from 1 to m; integrate with respect to t from 0 to Tn < T; to finally get: + ~-+-f~.k3u.,lu.,),+ f(t)tl+2. K bi(um)x,(u.~), t /=lo =~ro,fo f 1 K(um),F,. (t)[~ + 2, (9)

4 132 M E Kha//fa Integrate by parts to get: 1 1 K(u,,)z + -2 ~ K ~ ao(u,,,)x ' (u,,)x ~ l;r l_ t,12 ksu~ +2 t - Tt +B+2~or'~o~ t '+~+tl K(u.)t : , K au(u.)~,(u..)~, j=! + 2 Jo _.,t#+2----; 3u= f[,foi ff, fo,f. =1 = f(t)t-# + 2, KFs (u.)z - f(t)~ + 2, bi K(u.)x,(U,~)t -~:' fo l--l--- ~ ao(u.)xg~,(u.)t. (1O) t # + 2a ij = t Estimate the first term on the right hand side using the inequality labl ~< l/2a 2 + l/2b ~ and the theorem conditions to obtain: T, 1 Pu,,(u,)f + <~ Ifofo t # ~ 2~f(t) i f o'fo f(t)# 1 + 2, (u.)tr I I 1 f T'! p2 2 f:,fo 1 ~< 2 Jo f(t)t ~ + 2. u. + f(t)~ + z. (u.) 2 +i,f:,fo t-~ 1 "-~ q2. 01) From the above, it is easy to arrive at f(t)t#+2" u. 4(2 +/~+2a) t #+T'+ t where C is a constant independent of m, E. Similarly, we can estimate the remaining terms in (10), and therefore the inequality (10) can be written as: r, A (u=),' + B 2 r 1 tl+p+2, tl +,~+z. (urn)x,+ ~ u z t= t tl ~" P "*" 2dr ~< f(t)?+ 2, q 2, (13) where A=tl+2ut cnt cnt t CT~ 2 2# 2 ~b 8(2 +#+2a)'

5 Hyperbolic equation with discontinuous coefficients 133 # + 2~ Ct Ct and B = - - ). 2 2~ 2' choose T, sufficiently small such that A and B remain positive. From the integral on the right side in (13), together with the condition on q(x, t) we find (see [4]): f[,/, 1 tl +2. (u.),2 <C, t' +l (u.)~ < C, =1 /Or'yD t I +Jl+2a 1 u2<c. m (14) Multiply (8) by l/f (t); diffetientiate with respect to t and multiply the result by C~,, 1/tP; sum over s from 1 to m and integrate with respect to t from 0 to TI < T to get: +,-:k~(u..),(u.). + Jo 1,., + -~ % (K)x,(u.),x, (u=). f:fo' i/=1 if(t) " r, f() - ~ KF, (u,,),. (15) Io Io " From (15), the fact that IF.[ ~< IFI and the theorem conditions, we get the next a priori estimates: I:'ll) I~ (u-)2.<c, Io'Io I (u.),~, <, c, f:'~o I (u.)~ < C. (16) Using the a priori estimates (14) and (16) we can choose subsequences (u.a). (u=a)x,, (u,,a)., (u.a)zx, converges to u,, uz~, u., utx ' in Lz. Regarding the embedding theorem (see [1]) u., converges strongly to u.

6 134 M E Khalifa To prove the uniqueness, assume u and ~ are two different solutions and w -- u - 6. Using the a priori estimates in (14) and (16), we clearly get w = 0, i.e., u = ~. Thus, the existence and uniqueness have been proved. REMARK. Similarly, we can prove the existence theorem for problems (1)-(4), in case of F(x, t, u) = a(x, t)lut where ~ ~< n+2/n- 1 in place of condition 3 (see [10]). References [1] Adams R A, $oboi~ spaces, Academic Press, New York, [2] Chandirov G and Rzoev E, Approximation of solutions of nonregular mixed problems for equations of parabolic type with discontinuous and constant coefficients by solutions of regular problem, Az. Gos. Univ. Baku, USSR. (1981) [3] De Simon L, lnhomogeneous first order linear partial differential equations with coefficients and right hand side r ~ntinuous, Boll. Un. Mat. ltal. B (6) I (1982) [4] Duvaut G, Inequalities in mechanics and physics, Springer-Verla 8, Berlin, 1976 [5] Egarov E A, Boundary value problem for elliptic-parabolic equations, Sibirsk, Mat. Z. 17 (1976) [6] Hartman P, Ord/nary differential equations, John Wiley, New York, 1964 [7] John F and Klaiuerman S, Almost global ex~tence to nonlinear wave equation in three space dimensions, Commun. Pare Appl. Math., 37 (1984) [8] Klainerman S, Solutions to quasiliuear wave equations in three space dimensions, Commun. Pare Appi. Math. 36 (1983) [9] Krasnosclskii M A, Topolooieal methods in the theory of nonlinear integral equations, Macmillans, New York, 1964 [ 10] Nirenberg L, Variational and topological methods in nonlinear problems, Bull. Am. Math. Soc. 4 { 198 I) [11] Oleinik O A, Boundary value problem for linear parabolic equation with discontinuous coetficients, Uzv. Akad. Nauk SSR, So'. Mat., 25 (1961) 9-20 [12] Petrovski T G, Ord~u~ry differential equations, Prentice Hall, New Jersey, 1966 [13] Romnskevich L P, Classical solutions of second order quaailinear degenerate elliptic equations with discontinuous coelficlents, Usp. Mat. Nauk 36 (1980) [14] Shen Y T, Global boundeduess of weak solution of quasilinear elliptic equation of order two with strongly singular coefficients, J. China Univ. Sci. Tech. 11 (1981) [15] Sigillito V, Explicit a priori inequalities with applications to boundary value problem, Pitman, London, 1977 [16] Venturino M, The membership in H3(f~) of the solutions of a class of elliptic equations with discontinuous coefficients, Boll. Un. Mat. Ital. Suppl. (1980)