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 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 WORK, WE CONSIDER THE THERMOMECHANICAL PROBLEM DETERMINED BY A VISCOELASTIC PLATE COUPLED WITH HEAT CONDUCTION OF TYPE II. THIS PROBLEM DIFFERS SIGNIFICANTLY FROM WHAT CAN BE FOUND IN THE LITERATURE SINCE WE ASSUME THAT THE DISSIPATION MECHANISM IS MECHANICAL WHILE THE HEAT CONDUCTION MECHANISM IS CONSERVATIVE. WE FIRST PROVE THE EXISTENCE OF A SEMIGROUP OF CONTRACTIONS DEFINING THE SOLUTIONS IN A SUITABLE HILBERT SPACE. EXPONENTIAL STABILITY OF THE SOLUTIONS IS ALSO PROVED BY MEANS OF A KNOWN CHARACTERIZATION OF THE EXPONENTIALLY STABLE SEMIGROUPS. THE LACK OF DIFFERENTIABILITY OF THE SEMIGROUP IS ALSO PROVED. FINALLY, WE USE ENERGY ARGUMENTS TO SHOW THAT THE ONLY SOLUTION THAT CAN BE ZERO AFTER A FINITE TIME IS THE NULL SOLUTION.& COPY; 2023 ELSEVIER INC. ALL RIGHTS RESERVED.