IN THIS PAPER WE CONSIDER THE CLASSICAL LINEAR THEORY OF THERMOVISCOELASTICITY FOR INHOMOGENEOUS AND ANISOTROPIC MATERIALS IN THREE DIMENSIONAL SPACE. WE SHOW THAT UNDER SUITABLE CONDITIONS, THE SEMIGROUP ASSOCIATED WITH THE SYSTEM OF THE VISCOELASTIC EQUATION OF MOTION COUPLED WITH THE PARABOLIC EQUATION OF ENERGY IS ANALYTIC.

USING BOURGAIN SPACES AND THE GENERATOR OF DILATION P=3T ? T +X ? X , WHICH ALMOST COMMUTES WITH THE LINEAR KORTEWEG-DE VRIES OPERATOR, WE SHOW THAT A SOLUTION OF THE INITIAL VALUE PROBLEM ASSOCIATED FOR THE COUPLED SYSTEM OF EQUATIONS OF KORTEWEG-DE VRIES TYPE WHICH APPEARS AS A MODEL TO DESCRIBE THE STRONG INTERACTION OF WEAKLY NONLINEAR LONG WAVES, HAS AN ANALYTICITY IN TIME AND A SMOOTHING EFFECT UP TO REAL ANALYTICITY IF THE INITIAL DATA ONLY HAVE A SINGLE POINT SINGULARITY AT X=0.

THE PAPER ESTABLISHES THE EXPONENTIAL STABILITY OF A MICROPOLAR THERMOELASTIC HOMOGENEOUS AND LINEAR BODY WITH FRICTIONAL DISSIPATION AND THERMAL CONDUCTION OF HYPERBOLIC TYPE, ALLOWING FOR SECOND SOUND. EXISTENCE AND UNIQUENESS ARE PROVED WITHIN THE SEMIGROUP FRAMEWORK AND STABILITY IS ACHIEVED THANKS TO A SUITABLE LYAPUNOV FUNCTIONAL.

WE CONSIDER A LAMINATED BEAMS DUE INTERFACIAL SLIP WITH CONTROL BOUNDARY CONDITIONS OF FRACTIONAL DERIVATIVE TYPE. WE SHOW THE EXISTENCE AND UNIQUENESS OF SOLUTIONS. FURTHERMORE, CONCERNING THE ASYMPTOTIC BEHAVIOR WE SHOW THE LACK OF EXPONENTIAL STABILITY AND THE POLYNOMIAL DECAY RATE OF THE CORRESPONDING SEMIGROUP BY USING THE CLASSIC THEOREM OF BORICHEV AND TOMILOV.

WE STUDY ASYMPTOTIC BEHAVIOR OF THE SOLUTIONS OF A HYBRID SYSTEM WRAPPING AN EL LIPTIC OPERATOR

WE CONSIDER VIBRATIONS OF A NON UNIFORM FLEXIBLE STRUCTURE MODELED BY A D VISCOELASTIC EQUATION WITH KELVIN?VOIGT, COUPLED WITH AN EXPECTED DISSIPATIVE EFFECT : HEAT CONDUCTION GOVERNED BY CATTANEO?S LAW (SECOND SOUND). WE ESTABLISH THE WELL-POSEDNESS OF THE SYSTEM AND WE PROVE THE STABILIZATION TO BE EXPONENTIAL FOR ONE SET OF BOUNDARY CONDITIONS, AND AT LEAST POLYNOMIAL FOR ANOTHER SET OF BOUNDARY CONDITIONS. TWO DIFFERENT METHODS ARE USED: THE ENERGY METHOD AND ANOTHER MORE ORIGINAL, USING THE SEMIGROUP APPROACH AND STUDYING THE RESOLVENT OF THE SYSTEM.

THIS ARTICLE CONCERNS THE BEHAVIOUR OF SOLUTIONS TO A COUPLED SYSTEM OF SCHRODINGER EQUATIONS THAT HAS APPLICATIONS IN MANY PHYSICAL PROBLEMS, ESPECIALLY IN NONLINEAR OPTICS. IN PARTICULAR, WHEN THE SOLUTION EXISTS GLOBALLY, WE OBTAIN THE GROWTH OF THE SOLUTIONS IN THE ENERGY SPACE. FINALLY, SOME CONDITIONS ARE ALSO OBTAINED FOR HAVING BLOW-UP IN THIS SPACE.

IN THIS PAPER WE STUDY THE ASYMPTOTIC BEHAVIOR OF A EQUATION MODELING A MICROBEAM MOVING TRANSVERSALLY, COUPLED WITH AN EQUATION DESCRIBING A HEAT PULSE ON IT. SUCH PULSE IS GIVEN BY A TYPE III OF THE GREEN?NAGHDI MODEL, PROVIDING A MORE REALISTIC MODEL OF HEAT FLOW FROM A PHYSICS POINT OF VIEW. WE USE SEMIGROUPS THEORY TO PROVE EXISTENCE AND UNIQUENESS OF SOLUTIONS OF OUR MODEL, AND MULTIPLICATIVE TECHNIQUES TO PROVE EXPONENTIALLY STABLE OF ITS ASSOCIATED SEMIGROUP.

WE CONSIDER VIBRATIONS MODELED BY THE STANDARD LINEAR SOLID MODEL OF VISCOELASTICITY WHICH ARE COUPLED TO A HEAT EQUATION MODELING AN EXPECTEDLY DISSIPATIVE EFFECT THROUGH HEAT CONDUCTION. WE SHOW THAT THE EXPONENTIAL STABILITY UNDER THE FOURIER LAW OF HEAT CONDUCTION HOLDS. IN ORDER TO OBTAIN THE ASYMPTOTIC BEHAVIOUR WE USE MULTIPLIER TECHNIQUES.

WE STUDY THE EXISTENCE AND THE ASYMPTOTIC BEHAVIOR OF THE SOLUTION OF AN ABSTRACT VISCOELASTIC SYSTEM SUBMITTED TO NON-LOCAL INITIAL DATA. U(TT )+ AU - INTEGRAL(T)(0) G(T - S)BU(S)DS = 0 U(0) = XI(U) IN V, U(T) (0) = ETA(U) IN H, WHERE A AND B ARE DIFFERENTIAL OPERATORS SATISFYING B APPROXIMATE TO A(ALPHA) FOR 0

WE ANALYZE THE CONVERGENCE OF A NUMERICAL SCHEME FOR A CLASS OF DEGENERATE PARABOLIC PROBLEMS MODELLING REACTIONS IN POROUS MEDIA, AND INVOLVING A NONLINEAR, POSSIBLY VANISHING DIFFUSION. THE SCHEME INVOLVES THE KIRCHHOFF TRANSFORMATION OF THE REGULARIZED NONLINEARITY, AS WELL AS AN EULER IMPLICIT TIME STEPPING AND TRIANGLE BASED FINITE VOLUMES. WE PROVE THE CONVERGENCE OF THE APPROACH BY GIVING ERROR ESTIMATES IN TERMS OF THE DISCRETIZATION AND REGULARIZATION PARAMETER.

IN THIS PAPER WE STUDY THE ASYMPTOTIC BEHAVIOR OF A SYSTEM COMPOSED OF AN INTEGRO-PARTIAL DIFFERENTIAL EQUATION THAT MODELS THE LONGITUDINAL OSCILLATION OF A BEAM WITH A MEMORY EFFECT TO WHICH A THERMAL EFFECT HAS BEEN GIVEN BY THE GREEN-NAGHDI MODEL TYPE III, BEING PHYSICALLY MORE ACCURATE THAN THE FOURIER AND CATTANEO MODELS. TO ACHIEVE THIS GOAL, WE WILL USE ARGUMENTS FROM SPECTRAL THEORY, CONSIDERING A SUITABLE HYPOTHESIS OF SMOOTHNESS ON THE INTEGRO-PARTIAL DIFFERENTIAL EQUATION.

WE CONSIDER A COUPLED HYBRID SYSTEM WHOSE MAIN APPLICATION IS THE PROBLEM OF THE ACTIVE CONTROL OF NOISE. THE MODEL DESCRIBES THE INTERACTION OF ACOUSTIC VIBRATIONS IN THE INTERIOR OF A GIVEN TWO-DIMENSIONAL CAVITY WITH THE MECHANICAL VIBRATIONS OF TWO DAMPED STRINGS LOCATED IN A PART OF THE BOUNDARY OF THE CAVITY, IN WHICH SUITABLE FEEDBACKS ARE ACTING. OUR MAIN RESULT IS THAT THE TOTAL ENERGY ASSOCIATED TO THIS MODEL DECAYS EXPONENTIALLY AS TIME GOES TO INFINITY.

WE STUDY THE GAIN OF REGULARITY FOR THE INITIAL VALUE PROBLEM FOR A COUPLED NONLINEAR SCHRÖDINGER SYSTEM THAT DESCRIBES SOME PHYSICAL PHENOMENA SUCH AS THE PROPAGATION IN BIREFRINGENT OPTICAL FIBERS, KERR-LIKE PHOTO REFRACTIVE MEDIA IN OPTICS AND BOSE-EINSTEIN CONDENSATES. THIS STUDY IS MOTIVATED BY THE RESULTS OBTAINED BY N. HAYASHI ET AL.