WE CONSIDER THE TIMOSHENKO MODEL WITH PARTIAL DISSIPATIVE BOUNDARY CONDITION WITH DELAY, AND WE PROVE THAT THE SOLUTION DECAYS EXPONENTIALLY TO ZERO, PROVIDED THE WAVE SPEED ARE EQUAL; THIS IMPROVE EARLIER RESULT DUE TO BASSAM ET AL AND MUÑOZ RIVERA AND NASO. MOREOVER, CONSIDER THE EXPONENTIAL STABILITY TO THE CORRESPONDING SEMILINEAR PROBLEMS.

WE PROVE THE EXPONENTIAL STABILITY OF THE TIMOSHENKO BEAM WITH BOUNDARY DISSIPATION ONLY ON ONE SIDE OF THE BENDING MOMENT, PROVIDED THE WAVE SPEEDS OF THE SYSTEM ARE EQUAL.

WE CONSIDER THE THERMOELASTIC MODEL FOLLOWING THE TYPE III THEORY FOR THE EULER BERNOULLI BEAM EQUATION WITH TIP. WE PROVE THAT THE CORRESPONDING SEMIGROUP IS ANALYTIC. IN PARTICULAR, THIS IMPLIES: THE SMOOTHING EFFECT OVER THE INITIAL DATA, THE EXPONENTIAL STABILITY OF THE SEMIGROUP AND THAT THE RATE OF DECAY OF THE SEMIGROUP IS EQUAL TO THE SPECTRAL BOUND OF ITS GENERATOR (LINEAR STABILITY PROPERTY).

IN THIS PAPER, A THERMOELASTIC PROBLEM INVOLVING A VISCOUS STRAIN GRADIENT BEAM IS CONSIDERED FROM THE ANALYTICAL AND NUMERICAL POINTS OF VIEW. THE SO-CALLED TYPE II THERMAL LAW IS USED TO MODEL THE HEAT CONDUCTION AND TWO POSSIBLE DISSIPATION MECHANISMS ARE INTRODUCED IN THE MECHANICAL PART, WHICH IS CONSIDERED FOR THE FIRST TIME WITHIN STRAIN GRADIENT THEORY. AN EXISTENCE AND UNIQUENESS RESULT IS PROVED BY USING SEMIGROUP ARGUMENTS, AND THE EXPONENTIAL ENERGY DECAY IS OBTAINED. THE LACK OF DIFFERENTIABILITY FOR THE SEMIGROUP OF CONTRACTIONS IS ALSO SHOWN. THEN, FULLY DISCRETE APPROXIMATIONS ARE INTRODUCED BY USING THE FINITE ELEMENT METHOD AND THE BACKWARD TIME SCHEME, FOR WHICH A DEMONSTRATE THE ACCURACY OF THE APPROXIMATIONS AND THE BEHAVIOR OF THE DISCRETE ENERGY DECAY.

WE CONSIDER TWO SYSTEMS ARISING IN A TWO-DIMENSIONAL VISCOELASTIC FLUID-STRUCTURE INTERACTION, ONE OF THEM COUPLED WITH A WAVE EQUATION WITH INTERIOR DAMPING OF KELVIN?VOIGT TYPE. IN THE SECOND MODEL, THE ACOUSTIC VIBRATIONS OF THE VISCOUS FLUID WHICH FILLS THE TWO-DIMENSIONAL INTERIOR CAVITY ARE COUPLED WITH THE MECHANICAL VIBRATIONS OF A ONE-DIMENSIONAL THERMOELASTIC BEAM. THEREFORE, THE SYSTEMS ARE RELATED TO THE PROBLEM OF THE ACTIVE CONTROL OF NOISE IN A CAVITY. OUR MAIN RESULTS ARE THAT THE CORRESPONDING SEMIGROUPS ARE ANALYTIC. IN PARTICULAR WE SHOW THE EXPONENTIAL STABILITY OF THE MODELS.

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 CONSIDER THE UNIFORM STABILIZATION OF A HYBRID MODEL CONSISTING OF AN EULER?BERNOULLI BEAM WITH A TIP LOAD AT THE FREE END OF THE BEAM. OUR MAIN RESULT PROVES THAT THE SEMIGROUP ASSOCIATED WITH THIS MODEL IS NOT EXPONENTIALLY STABLE. ADDITIONALLY, WE PROVE THAT THE SEMIGROUP DECAYS POLYNOMIALLY TO ZERO AS T-1 AS TIME GOES TO INFINITY.

WE ANALYZE THE BRESSE SYSTEM WITH PARTIAL BOUNDARY DISSIPATION. OUR MAIN RESULT IS TO PROVE THAT THESE DISSIPATIVE MECHANISMS ARE ENOUGH TO STABILIZE EXPONENTIALLY THE WHOLE SYSTEM PROVIDED THE WAVE PROPAGATION SPEEDS ARE EQUAL.

WE STUDY THE EFFECT OF LOCALIZED VISCOELASTIC DISSIPATION FOR CURVED BEAMS. WE CONSIDER A CIRCULAR BEAM WITH THREE COMPONENTS, TWO OF THEM VISCOUS WITH CONSTITUTIVE LAWS OF KELVIN-VOIGT TYPE, ONE CONTINUOUS AND THE OTHER DISCONTINUOUS. THE THIRD COMPONENT IS ELASTIC WITHOUT ANY DISSIPATIVE MECHANISM. OUR MAIN RESULT IS THAT THE RATE OF DECAY DEPENDS ON THE POSITION OF EACH COMPONENT. MORE PRECISELY, WE PROVE THAT THE MODEL IS EXPONENTIALLY STABLE IF AND ONLY IF THE VISCOUS COMPONENT WITH DISCONTINUOUS CONSTITUTIVE LAW IS NOT IN THE CENTER OF THE BEAM. WE PROVE THAT WHEN THERE IS NO EXPONENTIAL STABILITY, THE SOLUTION DECAYS POLYNOMIALLY.

THE WELL-POSEDNESS AND STABILITY PROPERTIES OF A STOCHASTIC VISCOELASTIC EQUATION WITH MULTIPLICATIVE NOISE, LIPSCHITZ AND LOCALLY LIPSCHITZ NONLINEAR TERMS ARE INVESTIGATED. THE METHOD OF LYAPUNOV FUNCTIONS IS USED TO INVESTIGATE THE ASYMPTOTIC DYNAMICS WHEN ZERO IS NOT A SOLUTION OF THE EQUATION BY USING AN APPROPRIATE COCYCLE AND RANDOM DYNAMICAL SYSTEM. THE STABILITY OF MILD SOLUTIONS IS PROVED IN BOTH CASES OF LIPSCHITZ AND LOCALLY LIPSCHITZ NONLINEAR TERMS. FURTHERMORE, WE INVESTIGATE THE EXISTENCE OF A NON-TRIVIAL STATIONARY SOLUTION WHICH IS EXPONENTIALLY STABLE, BY USING A GENERAL RANDOM FIXED POINT THEOREM FOR GENERAL COCYCLES. IN THIS CASE, THE STATIONARY SOLUTION IS GENERATED BY THE COMPOSITION OF RANDOM VARIABLE AND WIENER SHIFT. IN ADDITION, THE THEORY OF RANDOM DYNAMICAL SYSTEM IS USED TO CONSTRUCT ANOTHER COCYCLE AND PROVE THE EXISTENCE OF A RANDOM FIXED POINT EXPONENTIALLY ATTRACTING EVERY PATH.

WE DEAL WITH THE SIGNORINI CONTACT PROBLEM BETWEEN TWO TIMOSHENKO BEAMS. IN THIS WORK WE USE THE THEORY OF SEMIGROUPS TO SHOW THE EXISTENCE OF SOLUTIONS THAT DECAY UNIFORMLY TO ZERO. THIS METHOD IS NEW AND MORE EFFECTIVE THAN THE WIDELY USED ENERGY METHOD. THIS IS BECAUSE IN PARTICULAR WE OBTAIN UNIFORM DECAY OF THE SOLUTIONS TO ZERO FOR ANY BOUNDARY CONDITION. A SECOND IMPORTANT POINT IS THAT WE CAN TAKE ADVANTAGE OF STABILIZATION RESULTS OF OTHERS LINEAR DYNAMIC SYSTEMS WITH DIFFERENT DISSIPATIVE MECHANISMS AND APPLY THEM THROUGH OUR METHOD FOR CONTACT PROBLEMS (SEE SECT. 4). FINALLY, THANKS TO LIPSCHITZIAN PERTURBATIONS WE CAN GENERALIZE THE SIGNORINI PROBLEM TO MORE GENERAL SEMI LINEAR PROBLEMS IN A SIMPLE WAY (SEE SECT. 4.3).

IN THIS PAPER, WE STUDY SIGNORINI?S PROBLEM FOR A LAMINATED TIMOSHENKO BEAM WITH INTERFACIAL SLIP. WE ASSUME THAT THE TRANSVERSE DISPLACEMENTS AT THE END OF THE BEAM ARE IN THE PRESENCE OF TWO RIGID OBSTACLES (SIGNORINI CONDITIONS). WE PROVE THE EXISTENCE OF SOLUTION AND ANALYZE THE ASYMPTOTIC BEHAVIOR OF THE SYSTEM. WE USE THE HYBRID-PENALIZED METHOD TO SHOW THE GLOBAL EXISTENCE OF AT LEAST ONE SOLUTION TO SIGNORINI?S PROBLEM AND FINALLY, NUMERICAL EXPERIMENTS VERIFY THE THEORETICAL RESULTS.

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.

IN THIS ARTICLE, WE USE THE EULER?BERNOULLI MODEL TO STUDY THE VIBRATIONS OF A BEAM COMPOSED OF TWO COMPONENTS, ONE CONSISTING OF A THERMOELASTIC MATERIAL AND THE OTHER OF A SIMPLY ELASTIC MATERIAL THAT DOES NOT PRODUCE DISSIPATION. OUR MAIN RESULT IS THAT THE SEMIGROUP ASSOCIATED WITH THIS MODEL IS DIFFERENTIABLE. IN PARTICULAR, OUR PROOF IMPLIES THE FOLLOWING PROPERTIES OF THE SEMIGROUP (1) IT IS OF GEVREY CLASS 12. (2) IT IS EXPONENTIALLY STABLE. (3) IT POSSESSES THE PROPERTY OF LINEAR STABILITY AND HAS A REGULARIZING EFFECT ON THE INITIAL DATA.

IN THIS ARTICLE WE STUDY THE BEHAVIOR OF THE SOLUTIONS FOR THE THREE-PHASE-LAG HEAT EQUATION WITH LOCALIZED DISSIPATION ON AN EULER-BERNOULLI BEAM MODEL. WE SHOW THAT SEMIGROUP S(T) ASSOCIATED WITH THE PROBLEM IS OF GEVREY CLASS 5 FOR T >0. IF THE COEFFICIENTS SATISFY TAU(ALPHA)>K & LOWAST;TAU(Q), THE SOLUTIONS ARE ALWAYS EXPONENTIALLY STABLE.

WE STUDY THE SEMIGROUP ASSOCIATED TO THE EULER--BERNOULLI BEAM EQUATION WITH LOCALIZED (DISCONTINUOUS) DISSIPATION. WE ASSUME THAT THE BEAM IS COMPOSED OF THREE COMPONENTS: ELASTIC, VISCOELASTIC OF KELVIN--VOIGT TYPE, AND THERMOELASTIC PARTS. WE PROVE THAT THIS MODEL GENERATES A SEMIGROUP OF GEVREY CLASS THAT IN PARTICULAR IMPLIES THE EXPONENTIAL STABILITY OF THE MODEL. TO OUR KNOWLEDGE, THIS IS THE FIRST POSITIVE RESULT GIVING INCREASED REGULARITY FOR THE EULER--BERNOULLI BEAM WITH LOCALIZED DAMPING.

WE CONSIDER THE EULER BERNOULLI BEAM MODEL WITH LOCALIZED THERMOELASTIC COMPONENT FOLLOWING THE GREEN AND NAGHDI TYPE III THEORY, SEE [6?9]. WE PROVE THAT THE CORRESPONDING SEMIGROUP IS OF GEVREY CLASS 8, REGARDLESS OF THE SIZE OF THE THERMOELASTIC COMPONENT OR THE POSITION IT OCCUPIES OVER THE BEAM. IN PARTICULAR, OUR RESULT IMPLIES THE INSTANTANEOUS SMOOTHING EFFECT PROPERTY OVER THE INITIAL DATA, THE EXPONENTIAL STABILITY OF THE SEMIGROUP AND THAT THE EXPONENTIAL RATE IS EQUAL TO THE SPECTRAL BOUND OF ITS INFINITESIMAL GENERATOR.

WE STUDY THE SEMI-LINEAR SIGNORINI PROBLEM FOR THE BRESSE BEAM (CURVED BEAMS) WITH FRICTIONAL DISSIPATION. WE SHOW THAT THERE EXISTS AT LEAST ONE WEAK SOLUTION THAT DECAYS EXPONENTIALLY TO ZERO WHEN THE SYSTEM IS TOTALLY DISSIPATIVE, OTHERWISE, IF THE DISSIPATIVE MECHANISMS ARE EFFECTIVE IN ONLY ONE OR TWO EQUATIONS OF THE SYSTEM, WE STILL SHOW THE EXPONENTIAL DECAY PROVIDED THE PROPAGATION SPEEDS OF THE MODEL ARE EQUAL TO EACH OTHER. IF THE SPEEDS OF PROPAGATIONS ARE DIFFERENT AND THE DISSIPATIVE MECHANISMS EFFECTIVE ONLY IN ONE OR TWO EQUATIONS THEN THE SOLUTION DECAYS POLYNOMIALLY IN GENERAL AS . THE EXCEPTION IS WHEN THE FRICTIONAL DISSIPATIVE MECHANISM IS EFFECTIVE ONLY OVER THE AXIAL FORCE IN WHICH CASE THE RATE OF DECAY IS . OUR MAIN TOOL IS THE SEMIGROUP THEORY APPLIED TO THE HYBRID MODEL THAT APPROXIMATES THE SIGNORINI PROBLEM. A NUMERICAL APPROACH IS PRESENTED TO HIGHLIGHT OUR THEORETICAL RESULTS.

WE CONSIDER THE HYBRID LAMINATED TIMOSHENKO BEAM MODEL. THIS STRUCTURE IS GIVEN BY TWO IDENTICAL LAYERS UNIFORM ON TOP OF EACH OTHER, TAKING INTO ACCOUNT THAT AN ADHESIVE OF SMALL THICKNESS IS BONDING THE TWO SURFACES AND PRODUCES AN INTERFACIAL SLIP. WE SUPPOSE THAT THE BEAM IS FASTENED SECURELY ON THE LEFT WHILE ON THE RIGHT IT?S FREE AND HAS AN ATTACHED CONTAINER. USING THE SEMIGROUP APPROACH AND A RESULT OF BORICHEV AND TOMILOV, WE PROVE THAT THE SOLUTION IS POLYNOMIALLY STABLE.