Global existence, uniform decay, and exponential growth of solutions for a system of viscoelastic Petrovsky equations

Abstract

In this paper, we study the initial-boundary value problem for a system of nonlinear viscoelastic Petrovsky equations. Introducing suitable perturbed energy functionals and using the potential well method we prove uniform decay of solution energy under some restrictions on the initial data and the relaxation functions. Moreover, we establish a growth result for certain solutions with positive initial energy.

Coupled system of nonlinear viscoelastic plate equations of (p(x),q(x))‐Kirchhoff‐type: Global existence, general decay, and blow‐up

“…They proved the global nonexistence of solutions with positive initial energy when the kernel of the memories satisfies some appropriate conditions. For more information about the results on the system of viscoelastic equations with constant exponents, we refer to the selected papers [5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20]. Related to the system of Kirchhoff-type equations, Li et al [21] considered the following system of viscoelastic wave equations of Kirchhoff-type with strong damping terms:…”

“…They proved the global nonexistence of solutions with positive initial energy when the kernel of the memories satisfies some appropriate conditions. For more information about the results on the system of viscoelastic equations with constant exponents, we refer to the selected papers [5–20].…”

Exponential Growth of Solutions for a Variable-Exponent Fourth-Order Viscoelastic Equation With Nonlinear Boundary Feedback

“…Song [28] considered equation (1.6) without strongly damping term and when (b = 0), he proved the nonexistence of solutions with positive initial energy and nonincreasing relaxation function. For more results related to the plate equations, we refer the reader to [3,29].…”

Global Nonexistence for a Nonlinear Viscoelastic Equation with Nonlinear Damping and Velocity-Dependent Material Density

“…However, they do not take the global nonexistence of the solution to (4) with nonlinear source into consideration. For more information about this, readers can refer to [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18] and references therein. It is well known that the nonlinear source term | | −2 causes global nonexistence of solutions when either the condition = 0 or ℎ( ) = 0 holds in (1)(see [5][6][7][8][9][10][11][12][13][14][15] and references cited therein).…”