| In recent years,as the development of mathematics and the rapid development of physics and mechanics,the study of nonlinear evolution equation has become one of the important issues in the partial differential equation research area.Especially, the study of the thermoelastic coupled rod ,beam and plane equations is a very active banch,which is focused by the academic researchers.However,the present results of the thermoelastic coupled equation are mainly about the existence and uniqueness of solutions,while the results of the global attractor existence and the dimension estimate are seldom.In this paper, we will make research about the global attractor existence and the dimension estimate of the thermoelastic coupled rod equationwhereΩ= (0, l),( l > 0), u = u ( x , t ) ,θ=θ( x ,t)are real functions defined onΩ×[ 0,+∞),which represent angular displacement from equilibrium and temperature difference to the reference temperature at position x and time t ,respectively.The details will go as follows: Firstly, the paper introduces the current domestic and international research situation and direction of the rod and beam equations briefly;Secondly, we put forward some important definitions and lemmas,and briefly explain some marks;Thirdly, we prove the existence and uniqueness of the weak solution,strong solution and classical solution for system (1)–(2) by means of the operator semigroups theory,and instruct the property that we can change the smoothness of solutions by changing the analyticity of the solution semigroups;Fourthly, we instruct the equivalent norm and prove the existence of the global attractor for solution semigroups and the dimension estimate;Fifthly, we make some prospects of the thermoelastic coupled evolution equation research in the future. |
