GLOBAL EXISTENCE FOR THE ONEDIMENSIONAL SEMILINEAR TRICOMITYPE EQUATIONS


 Edward O’Brien’
 1 years ago
 Views:
Transcription
1 GLOBAL EXISTENCE FOR THE ONEDIMENSIONAL SEMILINEAR TRICOMITYPE EQUATIONS ANAHIT GALSTYAN The Tricomi equation u tt tu xx = 0 is a linear partial differential operator of mixed type. (For t > 0, the Tricomi equation is hyperbolic; for t < 0, it is elliptic; while at t = 0, it has multiple characteristics.) In [4] and [5], one can find out about applications of the Tricomi equation. Although this equation is very well investigated, there is no result on the global in time, t 0, existence in the Cauchy problem for the nonlinear, even semilinear, Tricomi equation. It must be emphasized that a onedimensional wave equation essentially differs from the higher dimensional ones, x R n, u tt u = 0. In particular, Theorem 1 shows that there is no selfsimilar solution to a onedimensional wave equation, while for a three and higherdimensional wave equation, existence of such solutions recently is established (Pecher, Ribaud, Youssfi, Ozawa). (1) Glassey [2] studied the initial value problem, u(x, 0) = ψ 0 (x), u t (x, 0) = ψ 1 (x), x R n, for the semilinear wave equation (2) u tt u = u α+1, with n = 1, 2, 3. In particular, for the onedimensional wave equation, (3) u tt u xx = u α+1, u(x, 0) = ψ 0 (x), u t (x, 0) = ψ 1 (x), according to the results of [2], there is a finite time blow up for smooth compactly supported initial data if α > 0. Namely, if initial data are C smooth functions and have compact supports and positive averages, then for any α > 0, a classical solution of (2) cannot exist on R n [0, ). To motivate our approach consider integral equation (4) u(x, t) = u 0 (x, t) + G[ u α+1 ](x, t), corresponding to (3), where the function u 0 (x, t) = 1 ( ) ψ 0 (x + t) + ψ 0 (x t) x+t x t ψ 1 (y) dy
2 134 WOMEN IN MATHEMATICS: MAY 18 20, 2006 is the solution to the Cauchy problem for the linear string equation, while G[f](x, t) = 1 2 t 0 dτ t τ τ t f(x + z, τ)dz. If ψ 0, ψ 1 C 0, then u 0 C ([0, ) R), and u 0 (t, ) obeys the finite speed of propagation property. In particular, it has a compact support for every given instant t. Then, for every fixed T > 0 operator, G is continuous: G : C([0, T ] R) C([0, T ] R) and G : C([0, T ]; L p (R)) C([0, T ]; L q (R)). Any (distributional or classical) solution to the semilinear string equation also solves integral equation (4) with some function u 0 (t, x), which is a (distributional or classical) solution to the linear problem. On the other hand, if u 0 C 2 ([0, T ] R) is given, then any function u C([0, T ] R) which solves integral equation (4) is also solution of the semilinear equation. Denote K(x 0, t 0 ) := {(x, t) x x 0 < (t 0 t), 0 < t < t 0 }. The integral equation (4) is said to be obeying the Finite Speed of Propagation Property, if for every point (x 0, t 0 ) from u 0 (x, 0) = 0, t u 0 (x, 0) = 0 on {x R ; x x 0 t 0 } and from u 0 = 0 on K(x 0, t 0 ), it follows u = 0 on K(x 0, t 0 ) for the solution u C([0, T ]; L q (R)) C 1 ([0, T ]; D (R)). Theorem 1. Suppose that α > 0. For any given nontrivial function u 0 from C ([0, ) R) generated by ψ 0, ψ 1 C0 (R), ψ 1(x)dx 0, there is no global in time solution u C([0, ); L q (R)) C 1 ([0, T ]; D (R)) to the integral equation (4), which obeys the finite speed of propagation property. The next conjecture stresses the difference between onedimensional and higher dimensional semilinear wave equations. Conjecture (Strauss [7]). For n 2 blowup for all data if p < p n and global existence for all small data if p > p n. Here, n 2, p = α + 1, and p n is the positive root of the equation (n 1)p 2 n (n + 1)p n 2 = 0. For the history of the results that have validated Strauss s Conjecture, see [1]. In fact, the abovementioned difference between n = 1 and n 2 is more importunate. In particular, if we turn to nonsmooth data, for example, homogeneous, like ψ 0 (x) = x a, (a > 0) and ψ 1 (x) = x b, (b > 0), then it does not help to avoid a nonexistence. In the case of homogeneous initial data, the solution is selfsimilar: u(x, t) = λ 2/α u(λx, λt) for all λ > 0, x R, t > 0. Indeed, if we look for the selfsimilar solution to the Cauchy problem u tt u xx = u(x, t) α+1, u(x, 0) = 0, u t (x, 0) = ε x b, α > 0, then we have to set b = 1 + 2/α > 1. On the other hand, the solution u 0 = u 0 (x, t) of the Cauchy problem for the linear equation, (u 0 ) tt (u 0 ) xx = 0, u 0 (x, 0) = 0, (u 0 ) t (x, 0) = ε x b,
3 THE LEGACY OF LADYZHENSKAYA AND OLEINIK 135 is given by u 0 (x, t) = x+t x t s b /2 ds. In this case, since u(x, t) u 0 (x, t), the singularity of initial data is spread over the whole light cone and consequently, nonlinear onedimensional wave equation does not have selfsimilar solution. Wellposedness of the Cauchy problem for ndimensional linear Tricomi and Tricomitype equations was proved by many authors, among them O. A. Oleĭnik [6]. For the progress made in mixed type equations and transonic flow, see C. S. Morawetz [5] and B. L. Keyfitz [3]. In this paper, we establish global existence in the Cauchy problem for the semilinear onedimensional wave equation with the time dependent coefficient. Namely, consider the Cauchy problem (1) for the following equation, (5) u tt t 2k u xx = γ(t)f (u), with k 0, x R, t 0, a continuous function γ C((0, )), while F (u) is (6) F (u) = u α u or F (u) = u α+1, α > 0, u = u(x, t) is realvalued. We assume that for realvalued function γ, we have γ(t) Ct m for all t (0, ), with some constants C and m, m > 1. We study the Cauchy problem (5), (1) through integral equation. To write that integral equation, we appeal to [9, 3.3], where the following operator G is introduced: G[f](x, t) = (k + 1) k k k+1 t 0 x+φ(t) φ(b) db dy f(y, b) x (φ(t) φ(b)) (x y + φ(t) + φ(b)) γ (φ(b) (x y) + φ(t)) γ ( ) (x y + φ(t) φ(b))(x y φ(t) + φ(b)) F γ, γ; 1;, t 0. (x y + φ(t) + φ(b))(x y φ(t) φ(b)) Here, F ( a, b; c; z ) is the hypergeometric function, and φ(t) := tk+1 k + 1, γ := k 2 φ(1). As a matter of fact, operator G is a resolving operator for the Cauchy problem with zero initial data for the linear Tricomi equation, Now consider the integral equation u tt t 2k u xx = f(x, t). (7) u(x, t) = u 0 (x, t) + G[γ F (u)](x, t), x R, t 0, where function u 0 C([0, ); L q (R)) is given. The following theorem is the main result of the present communication.
4 136 WOMEN IN MATHEMATICS: MAY 18 20, 2006 Theorem 2. Assume that (8) q > 1, β(α + 1) m < 1, β = 2 + m α Let u 0 C([0, ); L q (R)) be given such that sup t β u 0 (t) L q (R) ε. (k + 1) q If ε is sufficiently small, then there exists a unique solution of equation (7) such that u C([0, ); L q (R)) sup t β u(t) L q (R) 2ε. If a given function u 0 C([0, ); L q (R)) solves the Cauchy problem for the linear equation (u 0 ) tt t 2k (u 0 ) xx = 0, u 0 (x, 0) = ψ 0 (x), u 0 t (x, 0) = ψ 1 (x), then we proved the following global existence theorem for the small data. Theorem 3. Assume that conditions (8), and inequalities 2 + m + α α(α + 1)(k + 1) 1 q < k 2α(k + 1), 1 q. 2 + m α(k + 1) are satisfied. Let ψ 0, ψ 1 be smooth functions with the compact supports, ψ 0, ψ 1 C0 (R) and small norms: ψ 0 L p (R) + ψ 1 L p (R) ε. Here p submits to terms 1 < p < ρ, 1/q = 1/p 1/ρ, 1/ρ + 1/ρ = 1. If ε is sufficiently small, then there exists a unique solution u C([0, ); L q (R)) C 1 ([0, ); D (R)) of the Cauchy problem (1), (5) such that sup t β u(t) L q (R) 2ε. According to the next theorem, the condition 0 < α < (2 + m)/k is necessary for the existence of the global in time weak solution to the problem (1), (5). It generalizes the statement of Theorem 1 to the Tricomitype equations. In the next theorem, the finite propagation speed property is defined by K k (x 0, t 0 ) := {(x, t) x x 0 < (t k+1 0 t k+1 )/(k+1), 0 < t < t 0 }. Theorem 4. Suppose that 0 < α < (2 + m)/k and 2k N. For any given nontrivial function u 0 C ([0, ) R) solution to the Cauchy problem for the linear Tricomitype equation, (u 0 ) tt t 2k (u 0 ) xx = 0,
5 THE LEGACY OF LADYZHENSKAYA AND OLEINIK 137 with the initial data ψ 0, ψ 1 C 0 (R), ψ 1(x)dx 0, there is no global in time solution u C([0, ); L q (R)) to the integral equation u(x, t) = u 0 (x, t) + G[ u α+1 ](x, t), which obeys the finite speed of propagation property. References [1] V. Georgiev, H. Lindblad, and C. D. Sogge, Weighted Strichartz estimates and global existence for semilinear wave equations, Amer. J. Math. 119 (1997), no. 6, [2] R. T. Glassey, Finitetime blowup for solutions of nonlinear wave equations, Math. Z. 177 (1981), no. 3, [3] B. L. Keyfitz, Courant Lecture in Honor of Cathleen Morawetz. NYU, November 20, [4] C. S. Morawetz, The mathematical approach to the sonic barrier, Bull. Amer. Math. Soc. (N.S.) 6 (1982), no. 2, [5], Mixed equations and transonic flow, J. Hyperbolic Differ. Equ. 1 (2004), no. 1, [6] O. A. Oleĭnik, On the Cauchy problem for weakly hyperbolic equations, Comm. Pure Appl. Math. 23 (1970), [7] W. A. Strauss, Nonlinear scattering theory at low energy, J. Funct. Anal. 41 (1981), no. 1, [8] F. Tricomi, Sulle equazioni lineari alle derivate parziali di secondo ordine di tipo misto, Rend. Reale Accad. Lincei Cl. Sci. Fis. Mat. Natur. 5 (1923), no. 14, [9] K. Yagdjian, A note on the fundamental solution for the Tricomitype equation in the hyperbolic domain, J. Differential Equations 206 (2004), no. 1, Department of Mathematics; University of TexasPan American; Edinburg, TX USA address:
DECAY ESTIMATES FOR THE KLEINGORDON EQUATION IN CURVED SPACETIME
Electronic Journal of Differential Equations, Vol. 218 218), No. 17, pp. 1 9. ISSN: 1726691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu DECAY ESTIMATES FOR THE KLEINGORDON EQUATION
More informationDISPERSIVE EQUATIONS: A SURVEY
DISPERSIVE EQUATIONS: A SURVEY GIGLIOLA STAFFILANI 1. Introduction These notes were written as a guideline for a short talk; hence, the references and the statements of the theorems are often not given
More informationarxiv: v2 [math.ap] 30 Jul 2012
Blow up for some semilinear wave equations in multispace dimensions Yi Zhou Wei Han. arxiv:17.536v [math.ap] 3 Jul 1 Abstract In this paper, we discuss a new nonlinear phenomenon. We find that in n space
More informationDETERMINATION OF THE BLOWUP RATE FOR THE SEMILINEAR WAVE EQUATION
DETERMINATION OF THE LOWUP RATE FOR THE SEMILINEAR WAVE EQUATION y FRANK MERLE and HATEM ZAAG Abstract. In this paper, we find the optimal blowup rate for the semilinear wave equation with a power nonlinearity.
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: 10726691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu ftp ejde.math.unt.edu (login:
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 informationLARGE TIME BEHAVIOR OF THE RELATIVISTIC VLASOV MAXWELL SYSTEM IN LOW SPACE DIMENSION
Differential Integral Equations Volume 3, Numbers 1 (1), 61 77 LARGE TIME BEHAVIOR OF THE RELATIVISTIC VLASOV MAXWELL SYSTEM IN LOW SPACE DIMENSION Robert Glassey Department of Mathematics, Indiana University
More informationMath 220A  Fall 2002 Homework 5 Solutions
Math 0A  Fall 00 Homework 5 Solutions. Consider the initialvalue problem for the hyperbolic equation u tt + u xt 0u xx 0 < x 0 u t (x, 0) ψ(x). Use energy methods to show that the domain of dependence
More informationPiecewise Smooth Solutions to the BurgersHilbert Equation
Piecewise Smooth Solutions to the BurgersHilbert Equation Alberto Bressan and Tianyou Zhang Department of Mathematics, Penn State University, University Park, Pa 68, USA emails: bressan@mathpsuedu, zhang
More informationA RIEMANN PROBLEM FOR THE ISENTROPIC GAS DYNAMICS EQUATIONS
A RIEMANN PROBLEM FOR THE ISENTROPIC GAS DYNAMICS EQUATIONS KATARINA JEGDIĆ, BARBARA LEE KEYFITZ, AND SUN CICA ČANIĆ We study a Riemann problem for the twodimensional isentropic gas dynamics equations
More informationA REMARK ON AN EQUATION OF WAVE MAPS TYPE WITH VARIABLE COEFFICIENTS
A REMARK ON AN EQUATION OF WAVE MAPS TYPE WITH VARIABLE COEFFICIENTS DANANDREI GEBA Abstract. We obtain a sharp local wellposedness result for an equation of wave maps type with variable coefficients.
More informationBLOWUP 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: 1726691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu BLOWUP OF SOLUTIONS
More informationMATH 220: Problem Set 3 Solutions
MATH 220: Problem Set 3 Solutions Problem 1. Let ψ C() be given by: 0, x < 1, 1 + x, 1 < x < 0, ψ(x) = 1 x, 0 < x < 1, 0, x > 1, so that it verifies ψ 0, ψ(x) = 0 if x 1 and ψ(x)dx = 1. Consider (ψ j )
More informationSimultaneous vs. non simultaneous blowup
Simultaneous vs. non simultaneous blowup Juan Pablo Pinasco and Julio D. Rossi Departamento de Matemática, F.C.E y N., UBA (428) Buenos Aires, Argentina. Abstract In this paper we study the possibility
More informationMathematical Research Letters 5, (1998) COUNTEREXAMPLES TO LOCAL EXISTENCE FOR QUASILINEAR WAVE EQUATIONS. Hans Lindblad
Mathematical Research Letters 5, 65 622 1998) COUNTEREXAMPLES TO LOCAL EXISTENCE FOR QUASILINEAR WAVE EQUATIONS Hans Lindblad 1. Introduction and themain argument In this paper, we study quasilinear wave
More informationMultiple positive solutions for a class of quasilinear elliptic boundaryvalue problems
Electronic Journal of Differential Equations, Vol. 20032003), No. 07, pp. 1 5. ISSN: 10726691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu login: ftp) Multiple positive
More informationarxiv: v1 [math.ap] 19 Mar 2011
LifeSan of Solutions to Critical Semilinear Wave Equations Yi Zhou Wei Han. Abstract arxiv:113.3758v1 [math.ap] 19 Mar 11 The final oen art of the famous Strauss conjecture on semilinear wave equations
More informationOn the Cauchy problem of 3D energycritical Schrödinger equations with subcritical perturbations
J. Differential Equations 30 (006 4 445 www.elsevier.com/locate/jde On the Cauchy problem of 3D energycritical Schrödinger equations with subcritical perturbations Xiaoyi Zhang Academy of Mathematics
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: 176691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu NONEXISTENCE OF GLOBAL SOLUTIONS OF CAUCHY PROBLEMS
More informationExact fundamental solutions
Journées Équations aux dérivées partielles SaintJeandeMonts, 5 juin 998 GDR 5 (CNRS) Exact fundamental solutions Richard Beals Abstract Exact fundamental solutions are known for operators of various
More informationBLOWUP 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: 7669. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu BLOWUP OF SOLUTIONS FOR VISCOELASTIC EQUATIONS OF KIRCHHOFF
More informationdoi: /j.jde
doi: 10.1016/j.jde.016.08.019 On Second Order Hyperbolic Equations with Coefficients Degenerating at Infinity and the Loss of Derivatives and Decays Tamotu Kinoshita Institute of Mathematics, University
More informationULAMHYERSRASSIAS STABILITY OF SEMILINEAR DIFFERENTIAL EQUATIONS WITH IMPULSES
Electronic Journal of Differential Equations, Vol. 213 (213), No. 172, pp. 1 8. ISSN: 1726691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu ULAMHYERSRASSIAS
More informationMathématiques appliquées (MATH05041) B. Dewals, Ch. Geuzaine
Lecture 2 The wave equation Mathématiques appliquées (MATH05041) B. Dewals, Ch. Geuzaine V1.0 28/09/2018 1 Learning objectives of this lecture Understand the fundamental properties of the wave equation
More informationUNIFORM DECAY OF SOLUTIONS FOR COUPLED VISCOELASTIC WAVE EQUATIONS
Electronic Journal of Differential Equations, Vol. 16 16, No. 7, pp. 1 11. ISSN: 176691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu UNIFORM DECAY OF SOLUTIONS
More informationOn Smoothness of Suitable Weak Solutions to the NavierStokes Equations
On Smoothness of Suitable Weak Solutions to the NavierStokes Equations G. Seregin, V. Šverák Dedicated to Vsevolod Alexeevich Solonnikov Abstract We prove two sufficient conditions for local regularity
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 informationNonlinear Control Systems
Nonlinear Control Systems António Pedro Aguiar pedro@isr.ist.utl.pt 3. Fundamental properties ISTDEEC PhD Course http://users.isr.ist.utl.pt/%7epedro/ncs2012/ 2012 1 Example Consider the system ẋ = f
More informationNONEXTINCTION OF SOLUTIONS TO A FAST DIFFUSION SYSTEM WITH NONLOCAL SOURCES
Electronic Journal of Differential Equations, Vol. 2016 (2016, No. 45, pp. 1 5. ISSN: 10726691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu NONEXTINCTION OF
More informationViscosity Iterative Approximating the Common Fixed Points of Nonexpansive Semigroups in Banach Spaces
Viscosity Iterative Approximating the Common Fixed Points of Nonexpansive Semigroups in Banach Spaces YUANHENG WANG Zhejiang Normal University Department of Mathematics Yingbing Road 688, 321004 Jinhua
More informationSimultaneous vs. non simultaneous blowup
Simultaneous vs. non simultaneous blowup Juan Pablo Pinasco and Julio D. Rossi Departamento de Matemática, F..E y N., UBA (428) Buenos Aires, Argentina. Abstract In this paper we study the possibility
More informationSharp blowup criteria for the DaveyStewartson system in R 3
Dynamics of PDE, Vol.8, No., 960, 011 Sharp blowup criteria for the DaveyStewartson system in R Jian Zhang Shihui Zhu Communicated by Y. Charles Li, received October 7, 010. Abstract. In this paper,
More informationarxiv:math/ v1 [math.ap] 28 Oct 2005
arxiv:math/050643v [math.ap] 28 Oct 2005 A remark on asymptotic completeness for the critical nonlinear KleinGordon equation Hans Lindblad and Avy Soffer University of California at San Diego and Rutgers
More informationOBSERVABILITY INEQUALITY AND DECAY RATE FOR WAVE EQUATIONS WITH NONLINEAR BOUNDARY CONDITIONS
Electronic Journal of Differential Equations, Vol. 27 (27, No. 6, pp. 2. ISSN: 72669. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu OBSERVABILITY INEQUALITY AND DECAY RATE FOR WAVE EQUATIONS
More informationCOMBINED EFFECTS FOR A STATIONARY PROBLEM WITH INDEFINITE NONLINEARITIES AND LACK OF COMPACTNESS
Dynamic Systems and Applications 22 (203) 37384 COMBINED EFFECTS FOR A STATIONARY PROBLEM WITH INDEFINITE NONLINEARITIES AND LACK OF COMPACTNESS VICENŢIU D. RĂDULESCU Simion Stoilow Mathematics Institute
More informationA NONLINEAR DIFFERENTIAL EQUATION INVOLVING REFLECTION OF THE ARGUMENT
ARCHIVUM MATHEMATICUM (BRNO) Tomus 40 (2004), 63 68 A NONLINEAR DIFFERENTIAL EQUATION INVOLVING REFLECTION OF THE ARGUMENT T. F. MA, E. S. MIRANDA AND M. B. DE SOUZA CORTES Abstract. We study the nonlinear
More informationGlobal Strichartz Estimates for Solutions of the Wave Equation Exterior to a Convex Obstacle
Global Strichartz Estimates for Solutions of the Wave Equation Exterior to a Convex Obstacle by Jason L. Metcalfe A dissertation submitted to the Johns Hopkins University in conformity with the requirements
More informationRegularity and compactness for the DiPerna Lions flow
Regularity and compactness for the DiPerna Lions flow Gianluca Crippa 1 and Camillo De Lellis 2 1 Scuola Normale Superiore, Piazza dei Cavalieri 7, 56126 Pisa, Italy. g.crippa@sns.it 2 Institut für Mathematik,
More informationA QuasiLinear Parabolic Partial Differential Equation with Accretive Property
ONLINE ISSN 8749 : Volume 3, Issue, 433438 A QuasiLinear Parabolic Partial Differential Equation with Accretive Property Aminu U. Bawa *, Micheal O. Egwurube and Murtala M. Ahmad 3 Department of Computer
More informationLIFE SPAN OF BLOWUP SOLUTIONS FOR HIGHERORDER SEMILINEAR PARABOLIC EQUATIONS
Electronic Journal of Differential Equations, Vol. 21(21), No. 17, pp. 1 9. ISSN: 1726691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu LIFE SPAN OF BLOWUP
More informationFOURIER INTEGRAL OPERATORS AND NONLINEAR WAVE EQUATIONS
MATHEMATICS OF GRAVITATION PART I, LORENTZIAN GEOMETRY AND EINSTEIN EQUATIONS BANACH CENTER PUBLICATIONS, VOLUME 41 INSTITUTE OF MATHEMATICS POLISH ACADEMY OF SCIENCES WARSZAWA 1997 FOURIER INTEGRAL OPERATORS
More informationFinitetime Blowup of Semilinear PDEs via the FeynmanKac Representation. CENTRO DE INVESTIGACIÓN EN MATEMÁTICAS GUANAJUATO, MEXICO
Finitetime Blowup of Semilinear PDEs via the FeynmanKac Representation JOSÉ ALFREDO LÓPEZMIMBELA CENTRO DE INVESTIGACIÓN EN MATEMÁTICAS GUANAJUATO, MEXICO jalfredo@cimat.mx Introduction and backgrownd
More informationSOME PROPERTIES ON THE CLOSED SUBSETS IN BANACH SPACES
ARCHIVUM MATHEMATICUM (BRNO) Tomus 42 (2006), 167 174 SOME PROPERTIES ON THE CLOSED SUBSETS IN BANACH SPACES ABDELHAKIM MAADEN AND ABDELKADER STOUTI Abstract. It is shown that under natural assumptions,
More informationSOLUTIONS OF SEMILINEAR WAVE EQUATION VIA STOCHASTIC CASCADES
Communications on Stochastic Analysis Vol. 4, No. 3 010) 45431 Serials Publications www.serialspublications.com SOLUTIONS OF SEMILINEAR WAVE EQUATION VIA STOCHASTIC CASCADES YURI BAKHTIN* AND CARL MUELLER
More informationSharp estimates for a class of hyperbolic pseudodifferential equations
Results in Math., 41 (2002), 361368. Sharp estimates for a class of hyperbolic pseudodifferential equations Michael Ruzhansky Abstract In this paper we consider the Cauchy problem for a class of hyperbolic
More informationarxiv: v4 [math.ap] 4 Jan 2017
Degeneracy in finite time of 1D quasilinear wave equations II Yuusuke Sugiyama Department of Mathematics, okyo University of Science Kagurazaka 13, Shinjukuku, okyo 1628601, Japan arxiv:1601.05191v4
More informationON A CONJECTURE OF P. PUCCI AND J. SERRIN
ON A CONJECTURE OF P. PUCCI AND J. SERRIN HansChristoph Grunau Received: AMSClassification 1991): 35J65, 35J40 We are interested in the critical behaviour of certain dimensions in the semilinear polyharmonic
More informationGlobal wellposedness for semilinear Wave and Schrödinger equations. Slim Ibrahim
Global wellposedness for semilinear Wave and Schrödinger equations Slim Ibrahim McMaster University, Hamilton ON University of Calgary, April 27th, 2006 1 1 Introduction Nonlinear Wave equation: ( 2
More informationGLOBAL WELLPOSEDNESS FOR NONLINEAR NONLOCAL CAUCHY PROBLEMS ARISING IN ELASTICITY
Electronic Journal of Differential Equations, Vol. 2017 (2017), No. 55, pp. 1 7. ISSN: 10726691. UL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu GLOBAL WELLPOSEDNESS FO NONLINEA NONLOCAL
More informationOn Asymptotic Variational Wave Equations
On Asymptotic Variational Wave Equations Alberto Bressan 1, Ping Zhang 2, and Yuxi Zheng 1 1 Department of Mathematics, Penn State University, PA 1682. Email: bressan@math.psu.edu; yzheng@math.psu.edu
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 informationSCATTERING FOR THE TWODIMENSIONAL NLS WITH EXPONENTIAL NONLINEARITY
SCATTERING FOR THE TWODIMENSIONAL NLS WITH EXPONENTIAL NONLINEARITY S. IBRAHIM, M. MAJDOUB, N. MASMOUDI, AND K. NAKANISHI Abstract. We investigate existence and asymptotic completeness of the wave operators
More informationDispersive Equations and Hyperbolic Orbits
Dispersive Equations and Hyperbolic Orbits H. Christianson Department of Mathematics University of California, Berkeley 4/16/07 The Johns Hopkins University Outline 1 Introduction 3 Applications 2 Main
More informationSELFSIMILAR SOLUTIONS FOR THE 2D BURGERS SYSTEM IN INFINITE SUBSONIC CHANNELS
Bull. Korean Math. oc. 47 010, No. 1, pp. 9 37 DOI 10.4134/BKM.010.47.1.09 ELFIMILAR OLUTION FOR THE D BURGER YTEM IN INFINITE UBONIC CHANNEL Kyungwoo ong Abstract. We establish the existence of weak
More informationL 1 Stability for scalar balance laws. Control of the continuity equation with a nonlocal flow.
L 1 Stability for scalar balance laws. Control of the continuity equation with a nonlocal flow. Magali Mercier Institut Camille Jordan, Lyon Beijing, 16th June 2010 Pedestrian traffic We consider tu +
More informationMATH 220 solution to homework 1
MATH solution to homework Problem. Define z(s = u( + s, y + s, then z (s = u u ( + s, y + s + y ( + s, y + s = e y, z( y = u( y, = f( y, u(, y = z( = z( y + y If we prescribe the data u(, = f(, then z
More informationNonlinear elliptic systems with exponential nonlinearities
22Fez 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 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: 10726691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu login: ftp) EXISTENCE
More informationApplication of wave packet transform to Schrödinger equations with a subquadratic potential
Application of wave packet transform to Schrödinger equations with a subquadratic potential Keiichi Kato(Tokyo University of Science) January 21, 2012 1 Introduction In this talk, we consider the following
More informationNONLINEAR DECAY AND SCATTERING OF SOLUTIONS TO A BRETHERTON EQUATION IN SEVERAL SPACE DIMENSIONS
Electronic Journal of Differential Equations, Vol. 5(5), No. 4, pp. 7. ISSN: 7669. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp) NONLINEAR DECAY
More informationGeneric Singularities of Solutions to some Nonlinear Wave Equations
Generic Singularities of Solutions to some Nonlinear Wave Equations Alberto Bressan Deartment of Mathematics, Penn State University (Oberwolfach, June 2016) Alberto Bressan (Penn State) generic singularities
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 672 99225872 Electronic Publishing House ON WEAK SOLUTION OF A HYPERBOLIC DIFFERENTIAL INCLUSION WITH NONMONOTONE DISCONTINUOUS NONLINEAR
More informationEXISTENCE RESULTS FOR OPERATOR EQUATIONS INVOLVING DUALITY MAPPINGS VIA THE MOUNTAIN PASS THEOREM
EXISTENCE RESULTS FOR OPERATOR EQUATIONS INVOLVING DUALITY MAPPINGS VIA THE MOUNTAIN PASS THEOREM JENICĂ CRÎNGANU We derive existence results for operator equations having the form J ϕu = N f u, by using
More informationON THE EXISTENCE AND NONEXISTENCE OF GLOBAL SIGN CHANGING SOLUTIONS ON RIEMANNIAN MANIFOLDS
Nonlinear Functional Analysis and Applications Vol. 2, No. 2 (25), pp. 2893 http://nfaa.kyungnam.ac.kr/journfaa.htm Copyright c 25 Kyungnam University Press KUPress ON THE EXISTENCE AND NONEXISTENCE
More informationAn Operator Theoretical Approach to Nonlocal Differential Equations
An Operator Theoretical Approach to Nonlocal Differential Equations Joshua Lee Padgett Department of Mathematics and Statistics Texas Tech University Analysis Seminar November 27, 2017 Joshua Lee Padgett
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: 1076691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu NONLINEAR SCHRÖDINGER
More informationNonlinear Schrödinger Equation BAOXIANG WANG. Talk at Tsinghua University 2012,3,16. School of Mathematical Sciences, Peking University.
Talk at Tsinghua University 2012,3,16 Nonlinear Schrödinger Equation BAOXIANG WANG School of Mathematical Sciences, Peking University 1 1 33 1. Schrödinger E. Schrödinger (18871961) E. Schrödinger, (1887,
More informationEQUADIFF 1. Rudolf Výborný On a certain extension of the maximum principle. Terms of use:
EQUADIFF 1 Rudolf Výborný On a certain extension of the maximum principle In: (ed.): Differential Equations and Their Applications, Proceedings of the Conference held in Prague in September 1962. Publishing
More informationExistence Results for Semipositone Boundary Value Problems at Resonance
Advances in Dynamical Systems and Applications ISSN 973531, Volume 13, Number 1, pp. 45 57 18) http://campus.mst.edu/adsa Existence Results for Semipositone Boundary Value Problems at Resonance Fulya
More informationNUMERICAL SIMULATIONS OF THE ENERGYSUPERCRITICAL NONLINEAR SCHRÖDINGER EQUATION
Journal of Hyperbolic Differential Equations Vol. 7, No. 2 (2010) 279 296 c World Scientific Publishing Company DOI: 10.1142/S0219891610002104 NUMERICAL SIMULATIONS OF THE ENERGYSUPERCRITICAL NONLINEAR
More informationEXISTENCE OF THREE WEAK SOLUTIONS FOR A QUASILINEAR DIRICHLET PROBLEM. Saeid Shokooh and Ghasem A. Afrouzi. 1. Introduction
MATEMATIČKI VESNIK MATEMATIQKI VESNIK 69 4 (217 271 28 December 217 research paper originalni nauqni rad EXISTENCE OF THREE WEAK SOLUTIONS FOR A QUASILINEAR DIRICHLET PROBLEM Saeid Shokooh and Ghasem A.
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: 1726691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ANALYSIS AND APPLICATION OF DIFFUSION EQUATIONS
More informationMULTIPLE SOLUTIONS FOR BIHARMONIC ELLIPTIC PROBLEMS WITH THE SECOND HESSIAN
Electronic Journal of Differential Equations, Vol 2016 (2016), No 289, pp 1 16 ISSN: 10726691 URL: http://ejdemathtxstateedu or http://ejdemathuntedu MULTIPLE SOLUTIONS FOR BIHARMONIC ELLIPTIC PROBLEMS
More informationFORCED OSCILLATIONS OF A CLASS OF NONLINEAR DISPERSIVE WAVE EQUATIONS AND THEIR STABILITY
Jrl Syst Sci & Complexity (2007) 20: 284 292 FORCED OSCILLATIONS OF A CLASS OF NONLINEAR DISPERSIVE WAVE EQUATIONS AND THEIR STABILITY Muhammad USMAN Bingyu ZHANG Received: 14 January 2007 Abstract It
More informationMATH 819 FALL We considered solutions of this equation on the domain Ū, where
MATH 89 FALL. The D linear wave equation weak solutions We have considered the initial value problem for the wave equation in one space dimension: (a) (b) (c) u tt u xx = f(x, t) u(x, ) = g(x), u t (x,
More informationNONHOMOGENEOUS ELLIPTIC EQUATIONS INVOLVING CRITICAL SOBOLEV EXPONENT AND WEIGHT
Electronic Journal of Differential Equations, Vol. 016 (016), No. 08, pp. 1 1. ISSN: 1076691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu NONHOMOGENEOUS ELLIPTIC
More informationNonstationary Friedrichs systems
Department of Mathematics, University of Osijek BCAM, Bilbao, November 2013 Joint work with Marko Erceg 1 Stationary Friedrichs systems Classical theory Abstract theory 2 3 Motivation Stationary Friedrichs
More informationMath 220a  Fall 2002 Homework 6 Solutions
Math a  Fall Homework 6 Solutions. Use the method of reflection to solve the initialboundary value problem on the interval < x < l, u tt c u xx = < x < l u(x, = < x < l u t (x, = x < x < l u(, t = =
More informationTADAHIRO OH 0, 3 8 (T R), (1.5) The result in [2] is in fact stated for timeperiodic functions: 0, 1 3 (T 2 ). (1.4)
PERIODIC L 4 STRICHARTZ ESTIMATE FOR KDV TADAHIRO OH 1. Introduction In [], Bourgain proved global wellposedness of the periodic KdV in L T): u t + u xxx + uu x 0, x, t) T R. 1.1) The key ingredient
More informationCONVERGENCE OF APPROXIMATING FIXED POINTS FOR MULTIVALUED NONSELFMAPPINGS IN BANACH SPACES. Jong Soo Jung. 1. Introduction
Korean J. Math. 16 (2008), No. 2, pp. 215 231 CONVERGENCE OF APPROXIMATING FIXED POINTS FOR MULTIVALUED NONSELFMAPPINGS IN BANACH SPACES Jong Soo Jung Abstract. Let E be a uniformly convex Banach space
More informationANALYTIC SMOOTHING EFFECT FOR NONLI TitleSCHRÖDINGER EQUATION IN TWO SPACE DIMENSIONS. Citation Osaka Journal of Mathematics.
ANALYTIC SMOOTHING EFFECT FOR NONLI TitleSCHRÖDINGER EQUATION IN TWO SPACE DIMENSIONS Author(s) Hoshino, Gaku; Ozawa, Tohru Citation Osaka Journal of Mathematics. 51(3) Issue 01407 Date Text Version publisher
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: 072669. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu EXISTENCE OF SOLUTIONS FOR KIRCHHOFF TYPE EQUATIONS
More informationSEMILINEAR ELLIPTIC EQUATIONS WITH DEPENDENCE ON THE GRADIENT
Electronic Journal of Differential Equations, Vol. 2012 (2012), No. 139, pp. 1 9. ISSN: 10726691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu SEMILINEAR ELLIPTIC
More informationCommon fixed points of generalized contractive multivalued mappings in cone metric spaces
MATHEMATICAL COMMUNICATIONS 365 Math. Commun., Vol. 14, No., pp. 365378 (009) Common fixed points of generalized contractive multivalued mappings in cone metric spaces Mujahid Abbas 1,, B. E. Rhoades
More informationGlobal existence of smooth solutions to twodimensional compressible isentropic Euler equations for Chaplygin gases
Global existence of smooth solutions to twodimensional compressible isentropic Euler equations for Chaplygin gases DeXing Kong a YuZhu Wang b a Center of Mathematical Sciences, Zhejiang University Hangzhou
More informationSingularity formation for compressible Euler equations
Singularity formation for compressible Euler equations Geng Chen Ronghua Pan Shengguo Zhu Abstract In this paper, for the psystem and full compressible Euler equations in one space dimension, we provide
More informationESTIMATES FOR ELLIPTIC HOMOGENIZATION PROBLEMS IN NONSMOOTH DOMAINS. Zhongwei Shen
W,p ESTIMATES FOR ELLIPTIC HOMOGENIZATION PROBLEMS IN NONSMOOTH DOMAINS Zhongwei Shen Abstract. Let L = div`a` x, > be a family of second order elliptic operators with real, symmetric coefficients on a
More informationQUANTUM MODELS FOR SEMICONDUCTORS AND NONLINEAR DIFFUSION EQUATIONS OF FOURTH ORDER
QUANTUM MODELS FOR SEMICONDUCTORS AND NONLINEAR DIFFUSION EQUATIONS OF FOURTH ORDER MARIA PIA GUALDANI The modern computer and telecommunication industry relies heavily on the use of semiconductor devices.
More informationSome asymptotic properties of solutions for Burgers equation in L p (R)
ARMA manuscript No. (will be inserted by the editor) Some asymptotic properties of solutions for Burgers equation in L p (R) PAULO R. ZINGANO Abstract We discuss time asymptotic properties of solutions
More informationStrauss conjecture for nontrapping obstacles
Chengbo Wang Joint work with: Hart Smith, Christopher Sogge Department of Mathematics Johns Hopkins University Baltimore, Maryland 21218 wangcbo@jhu.edu November 3, 2010 1 Problem and Background Problem
More informationThe Dirichlet problem for nondivergence parabolic equations with discontinuous in time coefficients in a wedge
The Dirichlet problem for nondivergence parabolic equations with discontinuous in time coefficients in a wedge Vladimir Kozlov (Linköping University, Sweden) 2010 joint work with A.Nazarov Lu t u a ij
More informationApplied Math Qualifying Exam 11 October Instructions: Work 2 out of 3 problems in each of the 3 parts for a total of 6 problems.
Printed Name: Signature: Applied Math Qualifying Exam 11 October 2014 Instructions: Work 2 out of 3 problems in each of the 3 parts for a total of 6 problems. 2 Part 1 (1) Let Ω be an open subset of R
More informationStructurally Stable Singularities for a Nonlinear Wave Equation
Structurally Stable Singularities for a Nonlinear Wave Equation Alberto Bressan, Tao Huang, and Fang Yu Department of Mathematics, Penn State University University Park, Pa. 1682, U.S.A. emails: bressan@math.psu.edu,
More informationPartial Differential Equations
Part II Partial Differential Equations Year 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2015 Paper 4, Section II 29E Partial Differential Equations 72 (a) Show that the Cauchy problem for u(x,
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/LMVUVSQ Joint work with Serge Nicaise and Cristina Pignotti Outline 1 Problem The idea Stability
More informationBulletin of the. Iranian Mathematical Society
ISSN: 1017060X (Print) ISSN: 17358515 (Online) Bulletin of the Iranian Mathematical Society Vol. 42 (2016), No. 1, pp. 129 141. Title: On nonlocal elliptic system of pkirchhofftype in Author(s): L.
More informationThe Concentrationcompactness/ Rigidity Method for Critical Dispersive and Wave Equations
The Concentrationcompactness/ Rigidity Method for Critical Dispersive and Wave Equations Carlos E. Kenig Supported in part by NSF 3 In these lectures I will describe a program (which I will call the
More informationSIMULTANEOUS AND NONSIMULTANEOUS BLOWUP AND UNIFORM BLOWUP PROFILES FOR REACTIONDIFFUSION SYSTEM
Electronic Journal of Differential Euations, Vol. 22 (22), No. 26, pp. 9. ISSN: 72669. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu SIMULTANEOUS AND NONSIMULTANEOUS
More informationPresenter: Noriyoshi Fukaya
Y. Martel, F. Merle, and T.P. Tsai, Stability and Asymptotic Stability in the Energy Space of the Sum of N Solitons for Subcritical gkdv Equations, Comm. Math. Phys. 31 (00), 347373. Presenter: Noriyoshi
More informationStrauss conjecture on asymptotically Euclidean manifolds
Strauss conjecture on asymptotically Euclidean manifolds Xin Yu (Joint with Chengbo Wang) Department of Mathematics, Johns Hopkins University Baltimore, Maryland 21218 xinyu@jhu.edu Mar 12Mar 13, 2010
More information