Wang, Peiguang

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Institute of Mathematics AS CR, v. v. i.

Mathematica SlovacaPei Guang WangOn the oscillation of solutions of parabolic partial functional differential equationsMathematica Slovaca, Vol. 51 (2001), No. 4, 449--458Persistent URL: http://dml.cz/dmlcz/131576Terms of use:© Mathematical Institute of the Slovak Academy of Sciences, 2001Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access todigitized documents strictly for personal use. Each copy of any part of this document must containthese Terms of use.This paper has been digitized, optimized for electronic delivery and stampedwith digital signature within the project DML-CZ: The Czech Digital MathematicsLibrary http://project.dml.czMathernatica Slovaca ©2001 . . . . . . . .--. /-.«,*.. \ _. _ . .« ..-.-. Mat hematical Inst i tute Ma tn . Slovaca, 5 1 ( 2 0 0 1 ) , No . 4 , 4 4 9 - 4 5 8 siovak Academy of Sciences ON THE OSCILLATION OF SOLUTIONS OF PARABOLIC PARTIAL FUNCTIONAL DIFFERENTIAL EQUATIONS PEIGUANG WANG (Communicated by Milan Medved?) ABSTRACT. Oscillation properties of the solutions of parabolic partial func-tional differential equations are investigated via the method of differential inequal-ities. The oscillatory criteria of solutions with three kinds of boundary conditions are obtained. 1. Introduction Recently there has been an increasing interest in studying parabolic differen-tial equations with functional arguments. In the oscillatory behavior of solutions for such equations some results have been received, we can refer to [1] - [4] and references therein. Those works, however, considered only the discrete deviating arguments. It seems that very little is known about the case with continuous de-viating arguments. The corresponding theory is not well-developed yet. In this paper, we consider the parabolic differential equation with continuous deviating arguments r\ — [u + X{t)u(x,r(t))] = a(t)Au - c (x , t , u ) - / q(x, t, £)u[x>#(*, 0] d ( 7 (0 + / (z , t), a (x, t) e íí x R+ = G, R+ = [0, + oo) (ì.i) 2000 M a t h e m a t i c s Sub jec t C l a s s i f i c a t i on : Primary 34C10, 34K11. Key words: oscillation, parabolic equation, distributed deviating arguments. Supported by Natural Science Foundation of Hebei Province of P. R. China. 449 PEIGUANG WANG and consider boundary value conditions of the type u = ip(x,t) on (z,£) G <9H x R + , (1.2a) Ou — =ij)(x,t) on (x,t)edttxR. , (1.2b) on i" du — + v(x,t)u = 0 on (x,t) G dVt x R, , (1.2c) cm "•" n 2 where At* --- £ J ^ r , u = u(x,t),ip(x,t),ip(x,t) G C(<9ft x R+, R), i/(x,t) G C(9fi x R+, R + ) , £} is a bounded domain in Rn with a piecewise smooth bound-ary dft, n denotes the unit exterior normal vector to dft. The aim of this paper is to establish oscillatory criteria for equation (1.1) satisfying three kinds of boundary conditions. Our approach is to reduce the multi-dimensional problem under study to a one-dimensional oscillation problem for ordinary differential equations or inequalities. Suppose that the following conditions (H) hold. (HJ a(t), X(t) G C(R + ,R + ) ; q(x,t,£) G C(f2xR+ x[a,6],R+) ; f(x,t) G C(ft x R + , R ) ; (H2) T(t) G C(R, ,R) and lim T(t) = +oo; (H3) g(t, £) G C(R+ x [a, b], R) ; g(t, £) < t, f G [a, b]; g(t, £) are nondecreas-ing with respect to t and £, and lim min {#(£,£)} = +oo; t->+oo fE[a,b] (H4) c(x,t,u) G C ( $ l x R + xR ,R) ; c(x,t,u) > p(t)r(u), in which p(t) G C(R + ,R + ) , r(u) G C([a,6],R) ; r(tt) is a positive and convex function in (0, +oo), and c(x, t, —u) = —c(x, t, u); (H5) o-(£) G ([a, b],R) is nondecreasing, integral of equation (1.1) is Stieltjes integral. Remark. Since integral of equation (1.1) is Stieltjes integral, equation (1.1) includes the following equation — [u + X(t)u(x, T(t))] = a(t)Au - c(x, t,u)-^2 Qj(x> l)u[x> Vj(*)] + /(x> *) • j=1 DEFINITION. A solution u(x,t) of equation (1.1) is called oscillatory in the do-main G if for each positive number \x there exists a point (x0, t0) G fi x [\x, +co) such that the condition u(x0,t0) = 0 holds. 450 ON THE OSCILLATION OF SOLUTIONS OF PARABOLIC DIFFERENTIAL EQUATIONS 2. Oscillation criteria The smallest eigenvalue ctx of the following Dirichlet problem in the do-main ft Au + au = 0 in (x, t) G ft x R+ , (2.1) u = 0 on (x, t) edftx R+ (2.2) is positive and the corresponding eigenfunction $(x) is positive in x 6 fi. LEMMA 1. Suppose that u is a positive solution of problem (1.1) and (1.2a) on ft x [//, +oo), /x > 0, £/zen the function U(t) satisfying the following differential inequality ^ [!/(*) + X(t)U(r(t))] + aia(t)U(t) +p(t)r(U(t)) ь + a (2.3) fQ(t,OU[g(t,0]MO<m) has eventually positive solution fu(x,t)$(x) dx U(t)=n , (2.4) J $(x) dx ft in which Q(t,€) = m j n { g ( a : , t , 0 } ; жЄft 1 H(t) = [ &(x) dx -a(t) f cp(x,t)^^- do;+ f f(x,t)$(x) dx ft J -• dft ft P r o o f . Let u(x,t) be a positive solution of problem (1.1) and (1.2a) in ft x [//, +oo), /i > 0. By condition ( H 3 ) , there exists a, tx > fj, such that 9(t,€) > /̂ > (^»0 £ [^1?+°°) x [«, 6] and T(t) > /J>, t > fj,, t > t1, then w(x,<7(*,0) > 0> ( ^ U ) £ fix [^,+00) x [a, 6], t/(x,r(*)) > 0, (x,£) G n x [^ ,+co) . Multiplying both sides of equation (1.1) by $ (# ) , and integrating 451 PEIGUANG WANG with respect to x over the domain ft, we have _d d* / uФ(x) dx + X(t) / u(x,т(t))Ф(x) dx n ӣ a(t) / AuФ(x) dx - c(яľ,t,г*)Ф(x) dx n n ь j I í (*, *, c>[*, g(t, 0 ] *(*) da(0 dx + | / (* , t)Ф(x) dx , t > *!. (2.5) íì a Using the Green formula, we have / Au$(x) dx= ( $ ( x ) - ^ - u ^ ) du) + / uA$(x) dx = - íU(x,t)^^\ áLj-ax fuA$(x) dx, * > * x . an a Using condition (H4) and Jensen's inequality, we have r r /Ju$(x)dx\ / c(x,t,u)$(x) dx > p(t) / r(u)$(x) dx > p(t)r af / $(x) dx , n n \ o / n * > * ! (2.7) Let us notice that o J J q(x, t, 0 « [x, </(*, 0 ] Ф(-c) da(0 dx П a = / / <?(x, ť, 0 « [x, g(t, 0 ] Ф(x) dx d a ( 0 • (2.8) a íž From (2.5)-(2.8), we have 452 ON THE OSCILLATION OF SOLUTIONS OF PARABOLIC DIFFERENTIAL EQUATIONS _d dí / u$(x) dx + X(t) / u(x,r(t))$(x) dx + axa(t) / u$(x) dx + p(t)r(U(t))J$(x) dx + JQ(t,o(fu[x,g(t,0]$(x) dx) d*(Q < - a(t) J V—J^- du + I f(x,t)$(x) dx, t>tx. dn si From the above inequality, we know that U(t) is an eventually positive solution of inequality (2.3). • THEOREM 1. Suppose that the function U(t) satisfying the following differen-tial inequalities ft[U(t) + X(t)U(T(t))] + aia(t)U(t)+p(t)r(U(t)) j. (2.9) + J Q(t,OU[g(t,0}MO<H(t), a ft [U(t) + X(t)U(T(t))] + aia(t)U(t)+p(t)r(U(t)) b + a I Q(t, t)U [g(t, 0] da(0 < -H(t) (2.10) have no eventually positive solution, then every solution of (1.1) and (1.2a) is oscillatory in G. P r o o f . Assume to the contrary that there exists a nonoscillatory solution u(x,t) of problem (1.1) and (1.2a). If u(x,t) > 0, (x,t) e ft x [fj,, +oo), // > 0, then from Lemma 1 it follows that U(t) is an eventually positive solution of inequality (2.9), which contradicts the condition of Theorem 1. If u(x,t) < 0, (x,t) G f ix [ti, 4-co), \x > 0, let v(x,t) = —u(x,t), then v(x, t) is a positive solution of the problem r\ -[u + X(t)u(X,T(t))] дt i(t)Au - c(x, t,u)- q(x, t,Ç)u[x,g(t, £)] da(Q + f(x, t), a u = —cp(xђt) on (xђt) Є дӣ x E + 453 PEIGUANG WANG and satisfies _d d* / u$(x) dx + X(t) / u(x,T(t))$(x) dx +axa(t) / u$(x) dx n n •- n + p(t)r(U(t))J$(x) dx + jQ(t,OU u[x,g(t,0]*(x) dx\ da(0 < -a(t) / ( - V ) ^ p dw+ ff$(x)dx, t>tt therefore fv(x,t)$(x) dx V " > = " f * ( x ) d x ( - » ) n would be an eventually positive solution of inequality (2.10), which contradicts the conditions of the theorem as well. This completes the proof of Theorem 1. • COROLLARY 1. Suppose that the function U(t) satisfying the following differ­ential inequality b ^[U(t) + X{t)U(T(t))]+aia(t)UW da(0 a < / / ( x , W x ) d x n has no eventually positive solution, then every solution u(x,t) of the below prob­lem r\ -\U + X(t)u(x,T(t))] дt a(t)Au - c(x, t,u)- q(x,t,£)u[x,g(t,£)] da(ţ) + f(x, t), a u = 0 on (x, t) Є ðíì x R+ is oscillatory in G. 454 ON THE OSCILLATION OF SOLUTIONS OF PARABOLIC DIFFERENTIAL EQUATIONS THEOREM 2. Suppose that t f \ f d$(x) f liminf / — a(s) / cp(x,s)— duj + f(x,s)<fr(x) dx ti an n t f \ f d$(x) f limsup / — a(s) / ip(x, s)— du + f(x,s)$(x) dx t->+oo J I J on J ti an n for sufficiently large tx, then every solution 0/(1.1) and (1.2a) is oscillatory in G. P r o o f . Assume that there exist nonoscillatory solution u(x,t). If u(x,t) > 0, (x,t) e Cl x [fi, +oo), // > 0, then from Lemma 1 it follows that the func­tion U(t) defined in (2.4) is an eventually positive solution of the inequality (2.9). Then ft[U(t) + X(t)U{r{t))] ds = —oo, (2.12) ds = +oo, J (2.13) < / Ф(x) dx n -a(t) f (p(x, * ) ^ | ^ da; + f f(x, t )$(x) dx L do, a Integrating the above inequality in the segment [tx, t], we get U{t) + X{t)U(T{t)) * > * ! < л - 1 t d s . / $(x) dx / -a(s) / <p(x,s) du+ f(x,s)$(x) dx n •* u an n From (2.12), we have liminf j [U(t) + \(t)U(T(t))] = - c o , which contradicts the assumption that U(t) > 0. If u(x, t) < 0, (x, t) e ft x [fi, +oo), \i > 0, let v(x, t) = -u(x, t), then V(t) is an eventually positive solution of the inequality (2.12). From (2.13), we get t liminf / — a(s) / (f(x,s) du> + f(x,s)$(x) dx ds ti дӣ г = — limsup / — a(s) / cp(x, s)—^— du + f(x,s)Ф(x) dx ds = t->+oo J L J ŐП J -oo u an n (2.14) thus , it can b e proved by using a m e t h o d similar t o t h e above-mentioned one. This proves T h e o r e m 2. D 455 PEIGUANG WANG C O R O L L A R Y 2. Suppose that liminf / / f(x,s)$(x) dxds = — oo , t-+ + oo J J t lim sup / / f(x, s)Ф(x) dx ds = +00 , t->-f-oo J J (2-15) (2-16) ti ӣ for sufficiently large tx, then every solution u(x,t) of the below problem -[u + \(t)U(x,T(t))} b = a(t)Au - c(x, t,u)- J q(x, t,f)«[x,g(t, 0] da(£) + f(x, t), a u = 0 on (x, t) e dn x M+ is oscillatory in G. Analogously to Theorem 1 and Theorem 2, we have following Theorems . THEOREM 3 . Suppose that the function V(t) satisfying the following differen-tial inequalities b £[V(t) + \(t)V(T(t))]+p(t)r(V(t))+ [Q(t,OV[g(t,0] da(0<G(t), i (2-17) b -£[V(t) + \(t)V(T(t))] +p(t)r(V(t)) + f Q(t,QV[g(t,0] da($ < -G(t), M { (2-18) have no eventually positive solution, then every solution of (1.1) and (1.2b) is oscillatory in G, where G(t) = r -1 - 1 i-/ dx a(t) / iþ(x,t) ăui+ f(x,t) dx -• Í2 -• -• дӣ П THEOREM 4 . Suppose that t liminf / a(s) / iþ(x,s) du + / f(x,t) dx U дӣ n t limsup / a(s) / iþ(x,s) du; + / f(x,t) dx t-+ + oo J L J J ds = - o o , (2.19) ds = + o o , (2.20) 456 ON THE OSCILLATION OF SOLUTIONS OF PARABOLIC DIFFERENTIAL EQUATIONS for sufficiently large £_, then every solution of (1.1) and (1.2b) is oscillatory in G. THEOREM 5. Suppose that the function V(t) satisfying the following differen-tial inequality b _d d*1 0 [V(t) + A(i)Y(т(*))l + P(t)r(V(t)) + f Q(t, OV[g(t,0] da(0 < 0, (2.21) has no eventually positive solution, then every solution of (1.1) and (1.2c) is oscillatory in G. THEOREM 6. Suppose that t liminf / / f(x,s) dxds = - c o , (2.22) t->+oo J J u n t limsup / / f(x,s) dxds = +oo (2.23) t-*+oo J J ti n for sufficiently large tx, then every solution of (1.1) and (1.2c) is oscillatory in G. To conclude this paper, we give an example. E X A M P L E . Consider the following equation o d d2u f — [u + u(x, t — IT)] = -—r — u — 2 / u(x, t + f) d£ + el cosx(2 sin t — e~n cos t), dt dx* J^ ( 2 2 4 ) uu uu — (0,t) = 0 , _ - - ( f > t ) - - - e * S i n t . (2.25) It is easy to see that conditions (H) holds, and t / \a(s) / ip(x,s) dou + / / ( x , s) dx ds ti on n t = \ [e5 sin s — e~n cos 5] ds 0 = e*sint i - e _ 7 r ] -el costly + e_7rl + - + e_7r . Hence all the conditions of Theorem 4 hold. It follows from Theorem 4 that every solution of problem (2.24), (2.25) is oscillatory in (0, ^) x [0,+co). For example u(x,t) = el cos x sin t is one oscillatory solution. 457 PEIGUANG WANG REFERENCES [1] YOSHIDA, N . : Oscillation of nonlinear parabolic equations with functional arguments, Hiroshima Math. J. 16 (1986), 305-314. [2] MISHIEV, D. P.—BAINOV, D. D . : Oscillation of the solution of parabolic differential equations of neutral type, Appl. Math . Comput . 28 (1988), 97-111. [3] YOSHIDA, N . : Forced oscillation of solutions of parabollic equations, Bull. Austral . Math. Soc. 36 (1987), 289-294. [4] FU, X. L.—ZHANG, L. Q. : Forced oscillation for ceHain nonlinear delay parabolic equa-tions, J. Part ial Differential Equatios 8 (1995), 82-88. [5] VLADIMIROV, V. S.: Equations of Mathematical Physics, Nauka, Moscow, 1981. Received June 25, 1999 Department of Mathematics Hebei University Baoding, 071002 P. R. CHINA E-mail: pgwang@mail.hbu.edu.cn 458 

On the oscillation of solutions of parabolic partial functional differential equations

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On the oscillation of solutions of parabolic partial functional differential equations

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Institute of Mathematics AS CR, v. v. i.