Abstract
The spatial properties of solutions for a class of thermoelastic plate with biharmonic operator were studied. The energy method was used. We constructed an energy expression. A differential inequality which the energy expression was controlled by a second-order differential inequality is deduced. The Phragme´n-Lindelo¨f alternative results of the solutions were obtained by solving the inequality. These results show that the Saint-Venant principle is also valid for the hyperbolic–hyperbolic coupling equations. Our results can been seen as a version of symmetry in inequality for studying the Phragme´n-Lindelo¨f alternative results.
Highlights
Saint Venant principle points out that for any equilibrium force system on an elastic body, if its action point is limited to a given ball, the displacement and stress generated by the equilibrium force system at any point where the distance from the load is far greater than the radius of the ball can be ignored
Many papers in the literature dealt with the study of the Saint-Venant principle
We can see that Liu and Lin [19] studied the spatial properties for time-dependent stokes equation. They transformed the equation to a biharmonic equation and obtained the Phragmén-Lindel ö f results by using a second-order differential inequality
Summary
Saint Venant principle points out that for any equilibrium force system on an elastic body, if its action point is limited to a given ball, the displacement and stress generated by the equilibrium force system at any point where the distance from the load is far greater than the radius of the ball can be ignored. Many scholars have begun to study the Phragmén-Lindel ö f alternative results of solutions. We can see that Liu and Lin [19] studied the spatial properties for time-dependent stokes equation They transformed the equation to a biharmonic equation and obtained the Phragmén-Lindel ö f results by using a second-order differential inequality. We try to establish the Phragmén-Lindel ö f alternative results for the solutions of the biharmonic Equations (4) and (5) under conditions (6) and (7). In order to get the Phragmén-Lindel ö f alternative results, we must define an energy expression for the solutions. We suggest u and v are classical solutions of problems (4)–(7), and we define a function φ3 (z, t) as: ωρ t z exp(−ωη )(z − ξ )v2,η dAdη ρ exp(−ωt)(z − ξ )v2,t dx dη.
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