OBSERVABILITY INEQUALITY AND DECAY RATE FOR WAVE EQUATIONS WITH NONLINEAR BOUNDARY CONDITIONS
|
|
- Peter Stokes
- 5 years ago
- Views:
Transcription
1 Electronic Journal of Differential Equations, Vol. 27 (27, No. 6, pp. 2. ISSN: URL: or OBSERVABILITY INEQUALITY AND DECAY RATE FOR WAVE EQUATIONS WITH NONLINEAR BOUNDARY CONDITIONS YUAN GAO, JIN LIANG, TI-JUN XIAO Communicated by Jerome A Goldstein Abstract. We study a class of wave propagation problems concerning the nonlinearity of dynamic evolution for boundary material. We establish an observability inequality for the related linear system, and make a connection between the linear system and the original nonlinear coupled system. Also, we obtain the desired energy decay rate for the original nonlinear boundary value problem.. Introduction We are concerned with the nonlinear boundary value problem u tt (x, t = u(x, t, x, t > ; (. u(x, t =, x Γ, t > ; (.2 u t (x, t f(z t g(z = x, t > ; (.3 u ν = z t x, t > ; (.4 u(x, = u (x, u t (x, = u (x, x, z(x, = z (x, x ; (.5 where is the Laplacian operator, is a bounded domain in R n with a boundary Γ = Γ (disjoint, closed, and nonempty of class C 2, and f, g are given functions on R. For some similar systems with or without source terms in (., there exist several results about uniform decay rate of the solutions to these systems. For instance, [6, 9,, ] study the porous boundary condition with the interface described by u t f(xz t g(xz =, x, t > ; u ν ρ(u t = z t, x, t >, where ρ is a given function. In this paper, we focus on the investigation of the problem above concerning the nonlinearity of dynamic evolution for boundary material, which is always described by boundary displacement z. We allow for nonlinear damping f(z t and nonlinear potential g(z (f and g may depend on x also, which 2 Mathematics Subject Classification. 35L7, 35B35, 76Q5, 35L2. Key words and phrases. Nonlinear wave; coupled boundary equations; uniform decay rates; observability inequality. c 27 Texas State University. Submitted September 8, 26. Published July 4, 27.
2 2 YUAN GAO, JIN LIANG, TI-JUN XIAO EJDE-27/6 can be handled similarly in the boundary displacement equation (.3. Such nonlinearity enables our results to possess wide applicability. Since our system is a coupled system and we hope to control the whole coupled system by only using a single boundary damping, which is different from and much more complex than the case of single equations, we will make efforts to establish the observability of the related linear system, to find a useful connection between the linear system and the original nonlinear system, and finally to obtain the decay rate of the energy. We also would like to state that our idea is stimulated by the significant papers [, 2, 4, 6, 7, 8,, 3, 4]. We first present some notation, basic definitions and assumptions (cf., e.g., [, 8]. Throughout this paper, c, c i are as generic constants whose values may change from line to line. We make the following assumptions: (H there exists x R n such that m(x ν(x for x Γ, where m(x = x x and ν(x is the unit normal vector pointing to the exterior of. (H2 The function g C(R is monotone nondecreasing such that g( = ; the function f C (R satisfies f( = and inf s R f (s >, and there exists a continuous strictly increasing odd function ρ C([, ]; R, which is continuously differentiable in a neighbourhood of with ρ( = ρ ( =, such that c ρ( v f(v c 2 ρ ( v, v, a.e. on, c v f(v c 2 v, v, a.e. on. Moreover, g(s is locally Lipschitz continuous such that Also we define (.6 c v g(v c 2 v, v, a.e.. (.7 H(x := xρ( x, x [, r 2 ], (.8 r > being small enough such that H is strictly convex on [, r 2 ]. We define {Ĥ (y/y, if y (,, L(y :=, if y =. (.9 Here Ĥ := sup{xy Ĥ(x} x R stands for the convex conjugate function of Ĥ (the extension of H to R in which Ĥ(x = for x R \ [, r 2 ]. Moreover, we define a function Λ H on (, r 2 ] by and write ψ(x := Λ H (x := H(x xh (x, H (r 2 H (r 2 /x v 2 ( Λ H ((H (v dv, x H (r 2. Then, there exists δ > such that ψ is strictly increasing on [, δ]. Let V ( = {u(x H (, u Γ = },
3 EJDE-27/6 WAVE EQUATIONS WITH NONLINEAR BOUNDARY CONDITIONS 3 and define the inner products and norms on V (, L 2 (, and L 2 ( respectively as follows ( /2, ((u, v V = u(x v(xdx, u V = u (x dx 2 (u, v = u(xv(xdx, u = ( (u(x 2 dx /2, φ, ψ = φ(xψ(xdγ, φ Γ = ( (φ(x 2 dx /2. Clearly, the V is equivalent to the usual H norm. Define the finite energy space by where the norm on H is given by H := V ( L 2 ( L 2 (, (u, v, z 2 H = u 2 V v 2 z 2. Define the energy of system (.-(.5 by E(t := u 2 u 2 t dx zt 2 dγ G(zdΓ, 2 2 where G(x = x g(sds is the anti-derivative of g. 2. Main results and proofs Rewrite the system (.-(.5 as u u t u u t = u = A u t. (2. t z f ( u t Γ g(z z The action of the operator A is given by the matrix of operators that appears in (2.. The remaining boundary conditions are encoded in the domain of A, given by { u D(A = v H; v V (, u L 2 (, u ν z Γ } = f ( v Γ g(z. From (H2, one knows that f is strictly increasing, and its inverse function f is Lipschitz continuous. Thus, using the standard method of nonlinear monotone operators and the semigroup theory (cf. [3], we can obtain wellposedness of the system. To study the energy decay rates of (.-(.5, we first give an observability inequality of the following linear system, which has the same initial values as the original nonlinear system: P tt (x, t = P (x, t, x, t > ; (2.2 P (x, t =, x Γ, t > ; (2.3 P t (x, t M t (x, t M(x, t =, x, t > ; (2.4 P (x, t = M t, x, t > ; (2.5 ν P (x, = u (x, P t (x, = u (x, x ; (2.6
4 4 YUAN GAO, JIN LIANG, TI-JUN XIAO EJDE-27/6 M(x, = z (x, x. (2.7 Using the multiplier method (cf., e.g., [2, 4], we can prove the following observability inequality. Theorem 2. (Observability inequality. There is a constant T >, depending only on, such that for T T, there corresponds a positive constant C T satisfying T C T E p ( Mt 2 dx dt, (2.8 where E p (t := 2 is the energy of (2.2-(2.7. P 2 t P 2 dx 2 Proof. The proof is divided into the following 5 steps. M 2 dγ Step : Let ξ(t C (R be the cutoff function defined by, t [, T ] ξ(t = a C function with range in (,, t (, (T, T, t (, (T,, for (, T/2. Let h be a [C 2 ( ] n -vector field, which will be specified later. Then, multiplying (2.2 by h P, integrating in time and space and using the boundary condition, we obtain T ɛ = h P (P tt P dx dt = (h P, P t L 2 ( T ɛ ( h ] 2 P 2 dx dt T ɛ T ɛ J P 2 dx dt = (h P, P t L 2 ( T ɛ Γ T ɛ T ɛ T ɛ T ɛ T ɛ h 2 (P 2 t P 2 dx dt h P M t dγdt, where J := hi(x x j. By (H we can take h such that [ ( h 2 (P 2 t h 2 P 2 t h P M t dγdt h ν = on Γ, h 2 P 2 dx dt h ν 2 (P t 2 P 2 dγdt T ɛ J P 2 dx dt J = h i(x x j ρ I on,
5 EJDE-27/6 WAVE EQUATIONS WITH NONLINEAR BOUNDARY CONDITIONS 5 for some constant ρ >. Hence, T ɛ ρ P 2 dx dt Since T ɛ T ɛ J P 2 dx dt h P M t dγdt (h P, P t L2 ( we have T ɛ ρ T ɛ T ɛ P 2 = (M 2 t P τ 2, P 2 dx dt T ɛ T ɛ h 2 (P t 2 P 2 dx dt [ T ɛ C h Mt 2 dγdt h ν 2 (P 2 t P 2 dγdt h 2 (P 2 t P 2 dx dt. E p = Mt 2 dγ, P 2 t P τ 2 dγdt ] C h E p (, where := (, T, and C h is a positive constant depending on h. Write l.o.t(p, M := (P, P t, M C([,T ];H ɛ ( H ɛ ( H ɛ (, (2.9 for ɛ >. Multiplying (2.2 by P h, integrating in time and space, and using the boundary condition and Sobolev Trace Theory, we obtain T ɛ h(pt 2 P 2 dx dt = P t, P h H ɛ ( H ɛ ( T ɛ C ɛ M 2 t dγdt ɛ P hm t dγdt T ɛ T T ɛ ɛ P P ( h dx dt P 2 dx dt l.o.t(p, M. Let min{ h} = d >. Combining (2. and (2.9 gives T ɛ P 2 Pt 2 dx dt C ɛ,h { M 2 t dγdt T ɛ C h E p ( l.o.t(p, M. (Pt 2 P } τ 2 dγdt (2. (2.
6 6 YUAN GAO, JIN LIANG, TI-JUN XIAO EJDE-27/6 Using [2, Lemma 4] to estimate T P τ 2 dγdt in (2., we obtain T ɛ P 2 Pt 2 dx dt { C T,ɛ,h Mt 2 ξ 2 Pt 2 dγdt C h E p ( l.o.t(p, M. T ɛ } Pt 2 dγdt (2.2 Step 2: We estimate T Pt 2 dγdt ξ 2 Pt 2 dγdt. The boundary condition on shows that T ɛ Pt 2 dγdt ξ 2 Pt 2 dγdt 2 Mt 2 M 2 dγdt. By (2.2, we have T ɛ E p (tdt C T,ɛ,h,f (Mt 2 M 2 dγdt C h E p ( l.o.t(p, M. (2.3 From E p = Mt 2 dγ, it follows that [ ] (T 2 E p ( Mt 2 dγdt (T 2 E p (T T ɛ C T,ɛ,h,f E p dt (M 2 t M 2 dγdt C h E p ( l.o.t(p, M. (2.4 Step 3: We estimate M 2 dγdt. Multiplying (2.4 by M and integrating in time and space, we obtain = M(P t M t MdΓdt Σ = MP dγ t=t (M t P MM t M 2 dγdt. t= Hence M 2 dγdt = MM t dγdt M t P dγdt MP dγ t=t t= (2.5 ɛ M Σ 2 dγdt C ɛ Mt 2 dγdt l.o.t.(p, M, where ɛ is arbitrarily small. Combining this with (2.4, we obtain (T 2 C h E p ( C T,ɛ,h Mt 2 dγdt l.o.t(p, M. (2.6 Therefore, for T > T := 2 C h, we almost get (2.8 except for the lower-order terms l.o.t(p, M. Step 5: We claim that for T > T = max{t, 2diam(},
7 EJDE-27/6 WAVE EQUATIONS WITH NONLINEAR BOUNDARY CONDITIONS 7 there exists a constant C T > such that the solution of (2.2-(2.7 satisfies the inequality l.o.t(p, M C T M t 2 L 2 (Σ. (2.7 Suppose this is false. Then there exists a sequence (P ( n, P t ( n, M( n H, and a corresponding solution sequence (P n, P n t, M n of (2.2-(2.7 such that l.o.t(p n, M n = n, M n t 2 L 2 ( n. Thus, by (2.6, we see that (P ( n, P t ( n, M( n H is bounded for T large enough. Hence there is a subsequence, still denoted by such that (P ( n, P t ( n, M( n, (P (, P t (, M(, (P ( n, P t ( n, M( n (P (, P t (, M(, in H weakly. (2.8 Let (P, Pt, M be the solution corresponding to (P (, P t (, M(. Then from E p = Mt 2 dγ <, it follows that (P n, P n t, M n (P, P t, M, weak star in L (, T ; H. (2.9 Clearly, (P n, P n t, M n C(,T ;H is bounded by the wellposedness of the system. Let X := H ( L 2 ( L 2 (, B := H ɛ ( H ɛ ( H ɛ (, Y := H ɛ ( (H ( H ɛ (. We claim that X B compactly. Indeed, for all s, t R with s > t, for an arbitrary bounded set {ψ n } H s (, we can extend the domain of ψ n to ˆ, such that ψ n ˆ =. It is known that H(ˆ s is compactly embedded in H(ˆ. t Hence, there exists a ψ H(ˆ t such that ψ ni ψ H t (ˆ. Hence ψ n i ψ H t (. We also claim that (P n t, P n tt, M n t L 2 (,T ;Y C uniformly. Indeed, it suffices to estimate Ptt n L2 (,T ;(H (. By (2.2 and the boundary condition, we see that for all t (, T and u H (, P tt, u = P udx = M t udγ P udx (2.2 is meaningful. Hence P tt L (, T ; (H (. We deduce then by a classic compactness result (see [2] that Therefore, (P n, P n t, M n (P, P t, M in L (, T ; B strongly. (P, P t, M C([,T ];H ɛ ( H ɛ ( H ɛ ( =. (2.2
8 8 YUAN GAO, JIN LIANG, TI-JUN XIAO EJDE-27/6 On the other hand, by (2.8, we have Mt =. Differentiating (2.4 in time, we obtain Ptt Γ =. Let a(t, x = Ptt(t, x such that a tt = a, in (, T, a ν = ( P ν tt =, on Γ (, T, a =, on. Using Holmgren s Uniqueness Theorem [], with T > 2diam(, Then from a(t, x = P tt(t, x =, in (, T. P =, in, P P Γ =, ν =, we know that P =. So we obtain M = due to (2.4. Thus (P, M = (, contradicts (2.2. A combination of Steps -5 completes the proof. Next we show a connection between linear and nonlinear systems. Theorem 2.2. Assume that (u, u t, z and (P, P t, M are solutions of system (.- (.5 and (2.2-(2.7 respectively. Then Mt 2 dγdt C zt 2 f(z t 2 dγdt. (2.22 Proof. Set ξ = u P, η = z M. Then (ξ, ξ t, η is the solution of ξ tt (x, t = ξ(x, t, x, t > ; ξ(x, t = x Γ, t > ; ν ξ t (x, t f(z t M t g(z M = x, t > ; ξ ν (x, t = η t(x, t x, t > ; ξ(x, =, ξ t (x, =, x ; η(x, =, x. Multiplying (2.23 by ξ t, integrating in time and space, we obtain t 2 ξ 2 t dx dt 2 t ξ = ν ξ tdγdt = η(m t f(z t M g(zdγdt Γ = (z t M t [M t f(z t M g(z]dγdt. Take ɛ > small enough. (.7 implies that there exist c >, c 2 > such that c v g(v c 2 v, v ɛ, a.e.. (2.23 (2.24
9 EJDE-27/6 WAVE EQUATIONS WITH NONLINEAR BOUNDARY CONDITIONS 9 Assuming z > ɛ, we have, by (2.24 and z t f(z t dγ, t t 2 ξ 2 t dx dt Mt 2 M t f(z t z t M t dγdt 2 t max{ ηη t, c ηη t }dγdt Therefore t t t t t 2 ξ 2 t dx dt 2 {η t } {η t<} t M t f(z t z t M t dγdt {η t } {η t<} max{ ηη t, c 2 ηη t }dγdt. M 2 t dγdt max{ ( η2 2 t, c ( η2 2 t}dγdt max{ ( η2 2 t, c 2 ( η2 2 t}dγdt. Noting the initial values and using Young s inequality, we obtain t t 2 ξ 2 t dx dt Mt 2 dγdt 2 t, (2.25 c f(z t 2 zt 2 dγdt giving (2.22. Similarly, we obtain (2.22 for z < ɛ. Finally, choose ɛ small enough such that g(z cɛ and z ɛ. By (2.24 we have t t 2 ξ 2 t dx dt = (z t M t [M t f(z t M z z g(z]dγdt, 2 and t t 2 ξ 2 2 η2 2 t dx dt [ M 2 t z t M t M t f(z t zz t M t z z t g(z M t g(z]dγdt. By Young s inequality and Hölder s inequality, we obtain t t 2 ξ 2 η2 2 2 t dx dt Mt 2 dγdt t [ Mt 2 C( (zt 2 f(z t 2 ɛ 2 ]dγdt. (2.26 Since the constant in (2.25 dose not depend on ɛ, we can let ɛ in (2.26. Noticing the initial values, we then obtain (2.22.
10 YUAN GAO, JIN LIANG, TI-JUN XIAO EJDE-27/6 Theorem 2.3 (Decay rate. Suppose that lim x H (x Λ H (x =, and T is a time such that (2.8 holds. Then the energy of system (.-(.5 satisfies ( E(t C(T, E(L ψ ( t T, T for t large enough; moreover, if then we have E(t C(T, E((H ( lim sup x Λ H (x <, c t T, for t large enough. Here, C(T, E( is a positive constant depending on T and E(, and T > depends on T. Proof. Clearly, we see that G(zdΓ = z {z } g(sds z { z } c z 2 dγ. 2 Setting c = max(c,, we have Let w satisfy c 2 sdsdγ z g(sdsdγ {z } z c sdsdγ E( c E p (. (2.27 H (w(s = sw(s β, s [, βr2, where { E( E( } β > max c L(H (r 2, (2.28, c δ r is as in (.8, and δ > is a constant such that ψ is strictly increasing on [, δ]. Then the definition of L implies ( s w(s = L, s [, βr 2 β. (2.29 From the property of L, it follows that w is a strictly increasing function from [, βr 2 onto [,. Thus, by using the optimal-weight convexity method (cf. [, Lemma 2.], we deduce that w(e p ( zt 2 f(z t 2 dγdt c 3 T H (w(e p ( c 4 (w(e p ( z t f(z t dγdt.
11 EJDE-27/6 WAVE EQUATIONS WITH NONLINEAR BOUNDARY CONDITIONS This and Theorems 2. and 2.2 yield C T E p (w(e p ( T T w(e p ( Mt 2 dγdt Cw(E p ( zt 2 f(z t 2 dγdt Γ T c 3 H (w(e p ( c 6 (w(e p ( z t f(z t dγdt Σ E p (w(e p ( T c 5 c 6 (H (r 2 β z t f(z t dγdt, E( c L(H (r 2 where we used (2.29 and β > (2.27, we have ( CT c 5T E( β Thanks to β > T c5 C T, we set and deduce that [ E(T E( ρ T w( E( ] c Denoting E k := E(kT c β, we obtain c in the last inequality. From this and ( E( w E( E(T. c ρ T := ( CT T c 5 > (2.3 c β E E [ ρ T L (E ]. [ = E( ρ T L ( E( ] c β. From the invariance by time translation t kt for system (.-(.5 and (2.2-(2.7, we have E k E k [ ρ T L (E k ]. Because β > E( c δ, we can apply [, Theorem.5] to complete the proof. Remark 2.4. Under the assumptions of Theorem 2.3, we have ( L ψ ( t T, as t. T Moreover, by taking special f and g, we can see clearly the meaning of the decay rate (please see the examples in [, Section 4]. Acknowledgments. This work was supported partly by the National Natural Science Foundation of China (3795, 57229, and the Shanghai Key Laboratory for Contemporary Applied Mathematics. References [] F. Alabau-Boussouira, K. Ammari; Sharp energy estimates for nonlinearly locally damped PDEs via observability for the associated undamped system, J. Funct. Anal., 26 (8 (2, [2] G. Avalos, I. Lasiecka; Exact controllability of structural acoustic interactions, J. Math. Pures Appl., 82 (8 (23, [3] V. Barbu; Nonlinear differential equations of monotone types in Banach spaces, Springer, New York, 2.
12 2 YUAN GAO, JIN LIANG, TI-JUN XIAO EJDE-27/6 [4] J. T. Beale, S. I. Rosencrans; Acoustic boundary conditions, Bull. Amer. Math. Soc., 8 (6 (974, [5] C. L. Frota, J. A. Goldstein; Some nonlinear wave equations with acoustic boundary conditions, J. Differential Equations, 64 ( (2, [6] C. L. Frota, N. A. Larkin; Uniform stabilization for a hyperbolic equation with acoustic boundary conditions in simple connected domains, Progr. Nonlinear Differential Equations Appl. 66(26, [7] C. Gal, G. R. Goldstein, J. A. Goldstein; Oscillatory boundary conditions for acoustic wave equations, J. Evol. Equ., 3 (4 (23, [8] Y. Gao, J. Liang, T. J. Xiao; A new method to obtain uniform decay rates for damped wave equations with nonlinear acoustic boundary conditions, submitted. [9] P. J. Graber; Wave equation with porous nonlinear acoustic boundary conditions generates a well-posed dynamical system, Nonlinear Anal., 73(9 (2, [] P. J. Graber; Strong stability and uniform decay of solutions to a wave equation with semilinear porous acoustic boundary conditions, Nonlinear Anal., 74 (2, [] P. J. Graber, B. Said-Houari; On the wave equation with semilinear porous acoustic boundary conditions, J. Differential Equations, 252 (9 (22, [2] J. Simon; Compact sets in the space L p (, T ; B, Ann. Mat. Pura Appl., (4 48 (987, [3] T. J. Xiao, J. Liang; Nonautonomous semilinear second order evolution equations with generalized Wentzell boundary conditions, J. Differential Equations, 252 (22, [4] T. J. Xiao, J. Liang; Coupled second order semilinear evolution equations indirectly damped via memory effects, J. Differential Equations, 254(5 (23, Yuan Gao Shanghai Key Laboratory for Contemporary Applied Mathematics, School of Mathematical Sciences, Fudan University, Shanghai 2433, China address: gaoyuan2@fudan.edu.cn Jin Liang School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai 224, China address: jinliang@sjtu.edu.cn Ti-Jun Xiao (corresponding author Shanghai Key Laboratory for Contemporary Applied Mathematics, School of Mathematical Sciences, Fudan University, Shanghai 2433, China address: tjxiao@fudan.edu.cn
Nonlinear stabilization via a linear observability
via a linear observability Kaïs Ammari Department of Mathematics University of Monastir Joint work with Fathia Alabau-Boussouira Collocated feedback stabilization Outline 1 Introduction and main result
More informationELLIPTIC EQUATIONS WITH MEASURE DATA IN ORLICZ SPACES
Electronic Journal of Differential Equations, Vol. 2008(2008), No. 76, pp. 1 10. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp) ELLIPTIC
More informationExponential stability of abstract evolution equations with time delay feedback
Exponential stability of abstract evolution equations with time delay feedback Cristina Pignotti University of L Aquila Cortona, June 22, 2016 Cristina Pignotti (L Aquila) Abstract evolutions equations
More informationBLOW-UP OF SOLUTIONS FOR VISCOELASTIC EQUATIONS OF KIRCHHOFF TYPE WITH ARBITRARY POSITIVE INITIAL ENERGY
Electronic Journal of Differential Equations, Vol. 6 6, No. 33, pp. 8. ISSN: 7-669. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu BLOW-UP OF SOLUTIONS FOR VISCOELASTIC EQUATIONS OF KIRCHHOFF
More informationEXISTENCE OF SOLUTIONS FOR KIRCHHOFF TYPE EQUATIONS WITH UNBOUNDED POTENTIAL. 1. Introduction In this article, we consider the Kirchhoff type problem
Electronic Journal of Differential Equations, Vol. 207 (207), No. 84, pp. 2. ISSN: 072-669. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu EXISTENCE OF SOLUTIONS FOR KIRCHHOFF TYPE EQUATIONS
More informationSTABILIZATION OF THE WAVE EQUATION WITH VARIABLE COEFFICIENTS AND A DYNAMICAL BOUNDARY CONTROL
Electronic Journal of Differential Equations, Vol. 06 06), No. 7, pp. 0. ISSN: 07-669. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu STABILIZATION OF THE WAVE
More informationMULTIPLE SOLUTIONS FOR CRITICAL ELLIPTIC PROBLEMS WITH FRACTIONAL LAPLACIAN
Electronic Journal of Differential Equations, Vol. 016 (016), No. 97, pp. 1 11. ISSN: 107-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu MULTIPLE SOLUTIONS
More informationEXISTENCE OF SOLUTIONS TO ASYMPTOTICALLY PERIODIC SCHRÖDINGER EQUATIONS
Electronic Journal of Differential Equations, Vol. 017 (017), No. 15, pp. 1 7. ISSN: 107-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu EXISTENCE OF SOLUTIONS TO ASYMPTOTICALLY PERIODIC
More informationEXISTENCE OF NONTRIVIAL SOLUTIONS FOR A QUASILINEAR SCHRÖDINGER EQUATIONS WITH SIGN-CHANGING POTENTIAL
Electronic Journal of Differential Equations, Vol. 2014 (2014), No. 05, pp. 1 8. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu EXISTENCE OF NONTRIVIAL
More informationA COUNTEREXAMPLE TO AN ENDPOINT BILINEAR STRICHARTZ INEQUALITY TERENCE TAO. t L x (R R2 ) f L 2 x (R2 )
Electronic Journal of Differential Equations, Vol. 2006(2006), No. 5, pp. 6. ISSN: 072-669. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp) A COUNTEREXAMPLE
More informationL 1 stability of conservation laws for a traffic flow model
Electronic Journal of Differential Equations, Vol. 2001(2001), No. 14, pp. 1 18. ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu ftp ejde.math.unt.edu (login:
More informationSEMILINEAR ELLIPTIC EQUATIONS WITH DEPENDENCE ON THE GRADIENT
Electronic Journal of Differential Equations, Vol. 2012 (2012), No. 139, pp. 1 9. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu SEMILINEAR ELLIPTIC
More informationPERIODIC SOLUTIONS FOR NON-AUTONOMOUS SECOND-ORDER DIFFERENTIAL SYSTEMS WITH (q, p)-laplacian
Electronic Journal of Differential Euations, Vol. 24 (24), No. 64, pp. 3. ISSN: 72-669. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu PERIODIC SOLUIONS FOR NON-AUONOMOUS
More informationUNIFORM DECAY OF SOLUTIONS FOR COUPLED VISCOELASTIC WAVE EQUATIONS
Electronic Journal of Differential Equations, Vol. 16 16, No. 7, pp. 1 11. ISSN: 17-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu UNIFORM DECAY OF SOLUTIONS
More informationBoundary Stabilization of Coupled Plate Equations
Palestine Journal of Mathematics Vol. 2(2) (2013), 233 242 Palestine Polytechnic University-PPU 2013 Boundary Stabilization of Coupled Plate Equations Sabeur Mansouri Communicated by Kais Ammari MSC 2010:
More informationNONLINEAR FREDHOLM ALTERNATIVE FOR THE p-laplacian UNDER NONHOMOGENEOUS NEUMANN BOUNDARY CONDITION
Electronic Journal of Differential Equations, Vol. 2016 (2016), No. 210, pp. 1 7. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu NONLINEAR FREDHOLM ALTERNATIVE FOR THE p-laplacian
More informationWELL-POSEDNESS OF WEAK SOLUTIONS TO ELECTRORHEOLOGICAL FLUID EQUATIONS WITH DEGENERACY ON THE BOUNDARY
Electronic Journal of Differential Equations, Vol. 2017 (2017), No. 13, pp. 1 15. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu WELL-POSEDNESS OF WEAK SOLUTIONS TO ELECTRORHEOLOGICAL
More informationStability of nonlinear locally damped partial differential equations: the continuous and discretized problems. Part II
. Stability of nonlinear locally damped partial differential equations: the continuous and discretized problems. Part II Fatiha Alabau-Boussouira 1 Emmanuel Trélat 2 1 Univ. de Lorraine, LMAM 2 Univ. Paris
More informationExponential decay for the solution of semilinear viscoelastic wave equations with localized damping
Electronic Journal of Differential Equations, Vol. 22(22), No. 44, pp. 1 14. ISSN: 172-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu (login: ftp) Exponential decay
More informationSOLUTION OF AN INITIAL-VALUE PROBLEM FOR PARABOLIC EQUATIONS VIA MONOTONE OPERATOR METHODS
Electronic Journal of Differential Equations, Vol. 214 (214), No. 225, pp. 1 1. ISSN: 172-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu SOLUTION OF AN INITIAL-VALUE
More informationViscosity Iterative Approximating the Common Fixed Points of Non-expansive Semigroups in Banach Spaces
Viscosity Iterative Approximating the Common Fixed Points of Non-expansive Semigroups in Banach Spaces YUAN-HENG WANG Zhejiang Normal University Department of Mathematics Yingbing Road 688, 321004 Jinhua
More informationGENERATORS WITH INTERIOR DEGENERACY ON SPACES OF L 2 TYPE
Electronic Journal of Differential Equations, Vol. 22 (22), No. 89, pp. 3. ISSN: 72-669. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu GENERATORS WITH INTERIOR
More informationAW -Convergence and Well-Posedness of Non Convex Functions
Journal of Convex Analysis Volume 10 (2003), No. 2, 351 364 AW -Convergence Well-Posedness of Non Convex Functions Silvia Villa DIMA, Università di Genova, Via Dodecaneso 35, 16146 Genova, Italy villa@dima.unige.it
More informationLIFE SPAN OF BLOW-UP SOLUTIONS FOR HIGHER-ORDER SEMILINEAR PARABOLIC EQUATIONS
Electronic Journal of Differential Equations, Vol. 21(21), No. 17, pp. 1 9. ISSN: 172-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu LIFE SPAN OF BLOW-UP
More informationStability of nonlinear locally damped partial differential equations: the continuous and discretized problems. Part I
. Stability of nonlinear locally damped partial differential equations: the continuous and discretized problems. Part I Fatiha Alabau-Boussouira 1 Emmanuel Trélat 2 1 Univ. de Lorraine, LMAM 2 Univ. Paris
More informationSYMMETRY IN REARRANGEMENT OPTIMIZATION PROBLEMS
Electronic Journal of Differential Equations, Vol. 2009(2009), No. 149, pp. 1 10. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu SYMMETRY IN REARRANGEMENT
More informationStability of an abstract wave equation with delay and a Kelvin Voigt damping
Stability of an abstract wave equation with delay and a Kelvin Voigt damping University of Monastir/UPSAY/LMV-UVSQ Joint work with Serge Nicaise and Cristina Pignotti Outline 1 Problem The idea Stability
More informationExact controllability of the superlinear heat equation
Exact controllability of the superlinear heat equation Youjun Xu 1,2, Zhenhai Liu 1 1 School of Mathematical Sciences and Computing Technology, Central South University, Changsha, Hunan 410075, P R China
More informationA MIXED PROBLEM FOR SEMILINEAR WAVE EQUATIONS WITH ACOUSTIC BOUNDARY CONDITIONS IN DOMAINS WITH NON-LOCALLY REACTING BOUNDARY
Electronic Journal of Differential Equations, Vol. 2014 (2014), No. 243, pp. 1 14. IN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu A MIXED PROBLEM
More informationUNIQUENESS OF SELF-SIMILAR VERY SINGULAR SOLUTION FOR NON-NEWTONIAN POLYTROPIC FILTRATION EQUATIONS WITH GRADIENT ABSORPTION
Electronic Journal of Differential Equations, Vol. 2015 2015), No. 83, pp. 1 9. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu UNIQUENESS OF SELF-SIMILAR
More informationHAIYUN ZHOU, RAVI P. AGARWAL, YEOL JE CHO, AND YONG SOO KIM
Georgian Mathematical Journal Volume 9 (2002), Number 3, 591 600 NONEXPANSIVE MAPPINGS AND ITERATIVE METHODS IN UNIFORMLY CONVEX BANACH SPACES HAIYUN ZHOU, RAVI P. AGARWAL, YEOL JE CHO, AND YONG SOO KIM
More informationNONLINEAR SCHRÖDINGER ELLIPTIC SYSTEMS INVOLVING EXPONENTIAL CRITICAL GROWTH IN R Introduction
Electronic Journal of Differential Equations, Vol. 014 (014), No. 59, pp. 1 1. ISSN: 107-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu NONLINEAR SCHRÖDINGER
More informationBLOW-UP OF SOLUTIONS FOR A NONLINEAR WAVE EQUATION WITH NONNEGATIVE INITIAL ENERGY
Electronic Journal of Differential Equations, Vol. 213 (213, No. 115, pp. 1 8. ISSN: 172-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu BLOW-UP OF SOLUTIONS
More informationSufficient conditions for functions to form Riesz bases in L 2 and applications to nonlinear boundary-value problems
Electronic Journal of Differential Equations, Vol. 200(200), No. 74, pp. 0. ISSN: 072-669. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu (login: ftp) Sufficient conditions
More informationWEAK ASYMPTOTIC SOLUTION FOR A NON-STRICTLY HYPERBOLIC SYSTEM OF CONSERVATION LAWS-II
Electronic Journal of Differential Equations, Vol. 2016 (2016), No. 94, pp. 1 14. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu WEAK ASYMPTOTIC
More informationGlobal Solutions for a Nonlinear Wave Equation with the p-laplacian Operator
Global Solutions for a Nonlinear Wave Equation with the p-laplacian Operator Hongjun Gao Institute of Applied Physics and Computational Mathematics 188 Beijing, China To Fu Ma Departamento de Matemática
More informationEXISTENCE OF WEAK SOLUTIONS FOR A NONUNIFORMLY ELLIPTIC NONLINEAR SYSTEM IN R N. 1. Introduction We study the nonuniformly elliptic, nonlinear system
Electronic Journal of Differential Equations, Vol. 20082008), No. 119, pp. 1 10. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu login: ftp) EXISTENCE
More informationSHARP BOUNDARY TRACE INEQUALITIES. 1. Introduction
SHARP BOUNDARY TRACE INEQUALITIES GILES AUCHMUTY Abstract. This paper describes sharp inequalities for the trace of Sobolev functions on the boundary of a bounded region R N. The inequalities bound (semi-)norms
More informationNONEXISTENCE OF GLOBAL SOLUTIONS OF CAUCHY PROBLEMS FOR SYSTEMS OF SEMILINEAR HYPERBOLIC EQUATIONS WITH POSITIVE INITIAL ENERGY
Electronic Journal of Differential Equations, Vol. 17 (17), No. 11, pp. 1 1. ISSN: 17-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu NONEXISTENCE OF GLOBAL SOLUTIONS OF CAUCHY PROBLEMS
More informationANALYSIS AND APPLICATION OF DIFFUSION EQUATIONS INVOLVING A NEW FRACTIONAL DERIVATIVE WITHOUT SINGULAR KERNEL
Electronic Journal of Differential Equations, Vol. 217 (217), No. 289, pp. 1 6. ISSN: 172-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ANALYSIS AND APPLICATION OF DIFFUSION EQUATIONS
More informationON WEAK SOLUTION OF A HYPERBOLIC DIFFERENTIAL INCLUSION WITH NONMONOTONE DISCONTINUOUS NONLINEAR TERM
Internat. J. Math. & Math. Sci. Vol. 22, No. 3 (999 587 595 S 6-72 9922587-2 Electronic Publishing House ON WEAK SOLUTION OF A HYPERBOLIC DIFFERENTIAL INCLUSION WITH NONMONOTONE DISCONTINUOUS NONLINEAR
More informationFUNCTIONAL COMPRESSION-EXPANSION FIXED POINT THEOREM
Electronic Journal of Differential Equations, Vol. 28(28), No. 22, pp. 1 12. ISSN: 172-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp) FUNCTIONAL
More informationSYMMETRY OF POSITIVE SOLUTIONS OF SOME NONLINEAR EQUATIONS. M. Grossi S. Kesavan F. Pacella M. Ramaswamy. 1. Introduction
Topological Methods in Nonlinear Analysis Journal of the Juliusz Schauder Center Volume 12, 1998, 47 59 SYMMETRY OF POSITIVE SOLUTIONS OF SOME NONLINEAR EQUATIONS M. Grossi S. Kesavan F. Pacella M. Ramaswamy
More informationConservative Control Systems Described by the Schrödinger Equation
Conservative Control Systems Described by the Schrödinger Equation Salah E. Rebiai Abstract An important subclass of well-posed linear systems is formed by the conservative systems. A conservative system
More informationPOTENTIAL LANDESMAN-LAZER TYPE CONDITIONS AND. 1. Introduction We investigate the existence of solutions for the nonlinear boundary-value problem
Electronic Journal of Differential Equations, Vol. 25(25), No. 94, pp. 1 12. ISSN: 172-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp) POTENTIAL
More informationSome lecture notes for Math 6050E: PDEs, Fall 2016
Some lecture notes for Math 65E: PDEs, Fall 216 Tianling Jin December 1, 216 1 Variational methods We discuss an example of the use of variational methods in obtaining existence of solutions. Theorem 1.1.
More informationASYMPTOTIC THEORY FOR WEAKLY NON-LINEAR WAVE EQUATIONS IN SEMI-INFINITE DOMAINS
Electronic Journal of Differential Equations, Vol. 004(004), No. 07, pp. 8. ISSN: 07-669. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp) ASYMPTOTIC
More informationMULTIPLE SOLUTIONS FOR THE p-laplace EQUATION WITH NONLINEAR BOUNDARY CONDITIONS
Electronic Journal of Differential Equations, Vol. 2006(2006), No. 37, pp. 1 7. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp) MULTIPLE
More informationarxiv: v1 [math.ap] 24 Oct 2014
Multiple solutions for Kirchhoff equations under the partially sublinear case Xiaojing Feng School of Mathematical Sciences, Shanxi University, Taiyuan 030006, People s Republic of China arxiv:1410.7335v1
More informationExistence and exponential stability of the damped wave equation with a dynamic boundary condition and a delay term.
Existence and exponential stability of the damped wave equation with a dynamic boundary condition and a delay term. Stéphane Gerbi LAMA, Université de Savoie, Chambéry, France Jeudi 24 avril 2014 Joint
More informationJUNXIA MENG. 2. Preliminaries. 1/k. x = max x(t), t [0,T ] x (t), x k = x(t) dt) k
Electronic Journal of Differential Equations, Vol. 29(29), No. 39, pp. 1 7. ISSN: 172-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu POSIIVE PERIODIC SOLUIONS
More informationASYMMETRIC SUPERLINEAR PROBLEMS UNDER STRONG RESONANCE CONDITIONS
Electronic Journal of Differential Equations, Vol. 07 (07), No. 49, pp. 7. ISSN: 07-669. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ASYMMETRIC SUPERLINEAR PROBLEMS UNDER STRONG RESONANCE
More informationStrongly nonlinear parabolic initial-boundary value problems in Orlicz spaces
2002-Fez conference on Partial Differential Equations, Electronic Journal of Differential Equations, Conference 09, 2002, pp 203 220. http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu
More informationBLOWUP THEORY FOR THE CRITICAL NONLINEAR SCHRÖDINGER EQUATIONS REVISITED
BLOWUP THEORY FOR THE CRITICAL NONLINEAR SCHRÖDINGER EQUATIONS REVISITED TAOUFIK HMIDI AND SAHBI KERAANI Abstract. In this note we prove a refined version of compactness lemma adapted to the blowup analysis
More informationInterior feedback stabilization of wave equations with dynamic boundary delay
Interior feedback stabilization of wave equations with dynamic boundary delay Stéphane Gerbi LAMA, Université Savoie Mont-Blanc, Chambéry, France Journée d EDP, 1 er Juin 2016 Equipe EDP-Contrôle, Université
More informationNON-EXTINCTION OF SOLUTIONS TO A FAST DIFFUSION SYSTEM WITH NONLOCAL SOURCES
Electronic Journal of Differential Equations, Vol. 2016 (2016, No. 45, pp. 1 5. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu NON-EXTINCTION OF
More informationNonlinear elliptic systems with exponential nonlinearities
22-Fez conference on Partial Differential Equations, Electronic Journal of Differential Equations, Conference 9, 22, pp 139 147. http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu
More informationNONTRIVIAL SOLUTIONS FOR SUPERQUADRATIC NONAUTONOMOUS PERIODIC SYSTEMS. Shouchuan Hu Nikolas S. Papageorgiou. 1. Introduction
Topological Methods in Nonlinear Analysis Journal of the Juliusz Schauder Center Volume 34, 29, 327 338 NONTRIVIAL SOLUTIONS FOR SUPERQUADRATIC NONAUTONOMOUS PERIODIC SYSTEMS Shouchuan Hu Nikolas S. Papageorgiou
More informationEXISTENCE OF BOUNDED SOLUTIONS FOR NONLINEAR DEGENERATE ELLIPTIC EQUATIONS IN ORLICZ SPACES
Electronic Journal of Differential Equations, Vol 2007(2007, o 54, pp 1 13 ISS: 1072-6691 URL: http://ejdemathtxstateedu or http://ejdemathuntedu ftp ejdemathtxstateedu (login: ftp EXISTECE OF BOUDED SOLUTIOS
More informationTwo dimensional exterior mixed problem for semilinear damped wave equations
J. Math. Anal. Appl. 31 (25) 366 377 www.elsevier.com/locate/jmaa Two dimensional exterior mixed problem for semilinear damped wave equations Ryo Ikehata 1 Department of Mathematics, Graduate School of
More informationExistence of Solutions for a Class of p(x)-biharmonic Problems without (A-R) Type Conditions
International Journal of Mathematical Analysis Vol. 2, 208, no., 505-55 HIKARI Ltd, www.m-hikari.com https://doi.org/0.2988/ijma.208.886 Existence of Solutions for a Class of p(x)-biharmonic Problems without
More informationJournal of Inequalities in Pure and Applied Mathematics
Journal of Inequalities in Pure and Applied athematics http://jipam.vu.edu.au/ Volume 4, Issue 5, Article 98, 2003 ASYPTOTIC BEHAVIOUR OF SOE EQUATIONS IN ORLICZ SPACES D. ESKINE AND A. ELAHI DÉPARTEENT
More informationA Dirichlet problem in the strip
Electronic Journal of Differential Equations, Vol. 1996(1996), No. 10, pp. 1 9. ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp (login: ftp) 147.26.103.110 or 129.120.3.113
More informationMULTIPLE POSITIVE SOLUTIONS FOR KIRCHHOFF PROBLEMS WITH SIGN-CHANGING POTENTIAL
Electronic Journal of Differential Equations, Vol. 015 015), No. 0, pp. 1 10. ISSN: 107-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu MULTIPLE POSITIVE SOLUTIONS
More informationOn periodic solutions of superquadratic Hamiltonian systems
Electronic Journal of Differential Equations, Vol. 22(22), No. 8, pp. 1 12. ISSN: 172-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu (login: ftp) On periodic solutions
More informationMinimization problems on the Hardy-Sobolev inequality
manuscript No. (will be inserted by the editor) Minimization problems on the Hardy-Sobolev inequality Masato Hashizume Received: date / Accepted: date Abstract We study minimization problems on Hardy-Sobolev
More informationCUTOFF RESOLVENT ESTIMATES AND THE SEMILINEAR SCHRÖDINGER EQUATION
CUTOFF RESOLVENT ESTIMATES AND THE SEMILINEAR SCHRÖDINGER EQUATION HANS CHRISTIANSON Abstract. This paper shows how abstract resolvent estimates imply local smoothing for solutions to the Schrödinger equation.
More informationCONSEQUENCES OF TALENTI S INEQUALITY BECOMING EQUALITY. 1. Introduction
Electronic Journal of ifferential Equations, Vol. 2011 (2011), No. 165, pp. 1 8. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu CONSEQUENCES OF
More informationHOMOCLINIC SOLUTIONS FOR SECOND-ORDER NON-AUTONOMOUS HAMILTONIAN SYSTEMS WITHOUT GLOBAL AMBROSETTI-RABINOWITZ CONDITIONS
Electronic Journal of Differential Equations, Vol. 010010, No. 9, pp. 1 10. ISSN: 107-6691. UL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu HOMOCLINIC SOLUTIONS FO
More informationSOLUTIONS TO SINGULAR QUASILINEAR ELLIPTIC EQUATIONS ON BOUNDED DOMAINS
Electronic Journal of Differential Equations, Vol. 018 (018), No. 11, pp. 1 1. ISSN: 107-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu SOLUTIONS TO SINGULAR QUASILINEAR ELLIPTIC EQUATIONS
More informationSobolev Spaces. Chapter Hölder spaces
Chapter 2 Sobolev Spaces Sobolev spaces turn out often to be the proper setting in which to apply ideas of functional analysis to get information concerning partial differential equations. Here, we collect
More informationTHE EDDY CURRENT PROBLEM WITH TEMPERATURE DEPENDENT PERMEABILITY
Electronic Journal of Differential Equations, Vol. 003003, No. 91, pp. 1 5. ISSN: 107-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu login: ftp THE EDDY CURRENT PROBLEM
More informationBilinear spatial control of the velocity term in a Kirchhoff plate equation
Electronic Journal of Differential Equations, Vol. 1(1), No. 7, pp. 1 15. ISSN: 17-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu (login: ftp) Bilinear spatial control
More informationOn a Nonlocal Elliptic System of p-kirchhoff-type Under Neumann Boundary Condition
On a Nonlocal Elliptic System of p-kirchhoff-type Under Neumann Boundary Condition Francisco Julio S.A Corrêa,, UFCG - Unidade Acadêmica de Matemática e Estatística, 58.9-97 - Campina Grande - PB - Brazil
More informationNew York Journal of Mathematics. A Refinement of Ball s Theorem on Young Measures
New York Journal of Mathematics New York J. Math. 3 (1997) 48 53. A Refinement of Ball s Theorem on Young Measures Norbert Hungerbühler Abstract. For a sequence u j : R n R m generating the Young measure
More informationIterative algorithms based on the hybrid steepest descent method for the split feasibility problem
Available online at www.tjnsa.com J. Nonlinear Sci. Appl. 9 (206), 424 4225 Research Article Iterative algorithms based on the hybrid steepest descent method for the split feasibility problem Jong Soo
More informationA CHARACTERIZATION OF STRICT LOCAL MINIMIZERS OF ORDER ONE FOR STATIC MINMAX PROBLEMS IN THE PARAMETRIC CONSTRAINT CASE
Journal of Applied Analysis Vol. 6, No. 1 (2000), pp. 139 148 A CHARACTERIZATION OF STRICT LOCAL MINIMIZERS OF ORDER ONE FOR STATIC MINMAX PROBLEMS IN THE PARAMETRIC CONSTRAINT CASE A. W. A. TAHA Received
More informationNonlocal Cauchy problems for first-order multivalued differential equations
Electronic Journal of Differential Equations, Vol. 22(22), No. 47, pp. 1 9. ISSN: 172-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu (login: ftp) Nonlocal Cauchy
More informationWeak and strong convergence theorems of modified SP-iterations for generalized asymptotically quasi-nonexpansive mappings
Mathematica Moravica Vol. 20:1 (2016), 125 144 Weak and strong convergence theorems of modified SP-iterations for generalized asymptotically quasi-nonexpansive mappings G.S. Saluja Abstract. The aim of
More informationForchheimer Equations in Porous Media - Part III
Forchheimer Equations in Porous Media - Part III Luan Hoang, Akif Ibragimov Department of Mathematics and Statistics, Texas Tech niversity http://www.math.umn.edu/ lhoang/ luan.hoang@ttu.edu Applied Mathematics
More informationCONVERGENCE OF EXTERIOR SOLUTIONS TO RADIAL CAUCHY SOLUTIONS FOR 2 t U c 2 U = 0
Electronic Journal of Differential Equations, Vol. 206 (206), No. 266, pp. 6. ISSN: 072-669. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu CONVERGENCE OF EXTERIOR SOLUTIONS TO RADIAL CAUCHY
More informationEXISTENCE OF SOLUTIONS TO THE CAHN-HILLIARD/ALLEN-CAHN EQUATION WITH DEGENERATE MOBILITY
Electronic Journal of Differential Equations, Vol. 216 216), No. 329, pp. 1 22. ISSN: 172-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu EXISTENCE OF SOLUTIONS TO THE CAHN-HILLIARD/ALLEN-CAHN
More informationElliptic equations with one-sided critical growth
Electronic Journal of Differential Equations, Vol. 00(00), No. 89, pp. 1 1. ISSN: 107-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu (login: ftp) Elliptic equations
More informationLERAY LIONS DEGENERATED PROBLEM WITH GENERAL GROWTH CONDITION
2005-Oujda International Conference on Nonlinear Analysis. Electronic Journal of Differential Equations, Conference 14, 2006, pp. 73 81. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
More informationFIXED POINT THEOREMS AND CHARACTERIZATIONS OF METRIC COMPLETENESS. Tomonari Suzuki Wataru Takahashi. 1. Introduction
Topological Methods in Nonlinear Analysis Journal of the Juliusz Schauder Center Volume 8, 1996, 371 382 FIXED POINT THEOREMS AND CHARACTERIZATIONS OF METRIC COMPLETENESS Tomonari Suzuki Wataru Takahashi
More informationAN EXTENSION OF THE LAX-MILGRAM THEOREM AND ITS APPLICATION TO FRACTIONAL DIFFERENTIAL EQUATIONS
Electronic Journal of Differential Equations, Vol. 215 (215), No. 95, pp. 1 9. ISSN: 172-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu AN EXTENSION OF THE
More informationMultiple positive solutions for a class of quasilinear elliptic boundary-value problems
Electronic Journal of Differential Equations, Vol. 20032003), No. 07, pp. 1 5. ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu login: ftp) Multiple positive
More informationSELF-ADJOINTNESS OF SCHRÖDINGER-TYPE OPERATORS WITH SINGULAR POTENTIALS ON MANIFOLDS OF BOUNDED GEOMETRY
Electronic Journal of Differential Equations, Vol. 2003(2003), No.??, pp. 1 8. ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu (login: ftp) SELF-ADJOINTNESS
More informationDETERMINATION OF THE BLOW-UP RATE FOR THE SEMILINEAR WAVE EQUATION
DETERMINATION OF THE LOW-UP RATE FOR THE SEMILINEAR WAVE EQUATION y FRANK MERLE and HATEM ZAAG Abstract. In this paper, we find the optimal blow-up rate for the semilinear wave equation with a power nonlinearity.
More informationEXISTENCE AND UNIQUENESS OF SOLUTIONS FOR A SECOND-ORDER NONLINEAR HYPERBOLIC SYSTEM
Electronic Journal of Differential Equations, Vol. 211 (211), No. 78, pp. 1 11. ISSN: 172-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu EXISTENCE AND UNIQUENESS
More informationGLOBAL ATTRACTOR FOR A SEMILINEAR PARABOLIC EQUATION INVOLVING GRUSHIN OPERATOR
Electronic Journal of Differential Equations, Vol. 28(28), No. 32, pp. 1 11. ISSN: 172-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp) GLOBAL
More informationA TWO PARAMETERS AMBROSETTI PRODI PROBLEM*
PORTUGALIAE MATHEMATICA Vol. 53 Fasc. 3 1996 A TWO PARAMETERS AMBROSETTI PRODI PROBLEM* C. De Coster** and P. Habets 1 Introduction The study of the Ambrosetti Prodi problem has started with the paper
More informationLocal null controllability of the N-dimensional Navier-Stokes system with N-1 scalar controls in an arbitrary control domain
Local null controllability of the N-dimensional Navier-Stokes system with N-1 scalar controls in an arbitrary control domain Nicolás Carreño Université Pierre et Marie Curie-Paris 6 UMR 7598 Laboratoire
More informationON THE EXISTENCE OF THREE SOLUTIONS FOR QUASILINEAR ELLIPTIC PROBLEM. Paweł Goncerz
Opuscula Mathematica Vol. 32 No. 3 2012 http://dx.doi.org/10.7494/opmath.2012.32.3.473 ON THE EXISTENCE OF THREE SOLUTIONS FOR QUASILINEAR ELLIPTIC PROBLEM Paweł Goncerz Abstract. We consider a quasilinear
More informationTHEOREMS, ETC., FOR MATH 515
THEOREMS, ETC., FOR MATH 515 Proposition 1 (=comment on page 17). If A is an algebra, then any finite union or finite intersection of sets in A is also in A. Proposition 2 (=Proposition 1.1). For every
More informationThe Split Hierarchical Monotone Variational Inclusions Problems and Fixed Point Problems for Nonexpansive Semigroup
International Mathematical Forum, Vol. 11, 2016, no. 8, 395-408 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/imf.2016.6220 The Split Hierarchical Monotone Variational Inclusions Problems and
More informationRELATIONSHIP BETWEEN SOLUTIONS TO A QUASILINEAR ELLIPTIC EQUATION IN ORLICZ SPACES
Electronic Journal of Differential Equations, Vol. 2014 (2014), No. 265, pp. 1 10. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu RELATIONSHIP
More informationShih-sen Chang, Yeol Je Cho, and Haiyun Zhou
J. Korean Math. Soc. 38 (2001), No. 6, pp. 1245 1260 DEMI-CLOSED PRINCIPLE AND WEAK CONVERGENCE PROBLEMS FOR ASYMPTOTICALLY NONEXPANSIVE MAPPINGS Shih-sen Chang, Yeol Je Cho, and Haiyun Zhou Abstract.
More informationA HYPERBOLIC PROBLEM WITH NONLINEAR SECOND-ORDER BOUNDARY DAMPING. G. G. Doronin, N. A. Lar kin, & A. J. Souza
Electronic Journal of Differential Equations, Vol. 1998(1998), No. 28, pp. 1 1. ISSN: 172-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp 147.26.13.11 or 129.12.3.113 (login: ftp) A
More informationMAXIMIZATION AND MINIMIZATION PROBLEMS RELATED TO A p-laplacian EQUATION ON A MULTIPLY CONNECTED DOMAIN. N. Amiri and M.
TAIWANESE JOURNAL OF MATHEMATICS Vol. 19, No. 1, pp. 243-252, February 2015 DOI: 10.11650/tjm.19.2015.3873 This paper is available online at http://journal.taiwanmathsoc.org.tw MAXIMIZATION AND MINIMIZATION
More information