Blow-up for a viscoelastic von Karman equation with strong damping and variable exponent source terms

In this article, we deal with a strongly damped von Karman equation with variable exponent source and memory effects. We investigate blow-up results of solutions with three levels of initial energy such as non-positive initial energy, certain positive initial energy, and high initial energy. Furthermore, we estimate not only the upper bound but also the lower bound of the blow-up time.


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The Von Karman bracket [·, ·] is given as here x = (x 1 , x 2 ) ∈ . The exponent function q(·) is measurable and verifies for all x, x ∈ with |x -x| < κ, (1.5) where a > 0 and 0 < κ < 1, and 2 ≤ q 1 := ess inf h(ts) 2 w(s) ds + g(x, w t ) = w, χ(w) (1.6) associated with different boundary conditions has been intensively treated about existence and stability (see [5,8,9,[16][17][18] and the references therein). When h = g = 0, the authors of [8] studied the unique existence of global solution. When g(x, u t ) = a(x)u t and h = 0, Horn and Lasiecka [9] discussed energy decay estimates. When g = 0, Park et al. [18] obtained the general decay behavior of solutions. On the other hand, problems with variable exponent source have been attracting great interest [13][14][15]21]. Such problems appear in physical phenomena such as nonlinear elastics [22], electrorheological fluids [20], stationary thermorheological viscous flows of non-Newtonian fluids [2], and image precessing [1]. Recently, Messaoudi et al. [15] considered wave equations with source and damping terms of variable exponent, They obtained the local existence of solutions under appropriate conditions on γ (·) and q(·) by utilizing the Faedo-Galerkin's technique and the contraction mapping theorem. Furthermore, they showed that the solution with negative initial energy blows up in a finite time. Later, Park and Kang [19] improved and complemented the result of [15] by obtaining a blow-up result of solution with certain positive initial energy for a wave equation of memory type. We also refer to a recent work [4] for a nonlinear diffusion system involving variable exponents dependent on spatial and time variables and cross-diffusion terms. At this point, it is worthwhile to mention that there is little work concerning global nonexistence of solutions for viscoelastic von Karman equations with variable source effect. Particularly, there is no literature concerning blow-up results of solutions with high initial energy for the equations. Thus, in this article, we establish blow-up results of solutions with three levels of initial energy such as non-positive initial energy, certain positive initial energy, and high initial energy. Furthermore, we estimate not only the upper bound but also the lower bound of the blow-up time. These are inspired by the ideas of [23], where the authors proved blow-up results of solutions with high initial energy and estimated bounds of existence time of solutions for wave equation with logarithmic nonlinear source term.
The outline of this article is here. In Sect. 2, we state some definitions, notations, and auxiliary lemmas. In Sect. 3, we construct blow-up results and obtain bounds of the blowup time.

Preliminaries
We denote (y, v) = y(x)v(x) dx, y 2 2 = (y, y). Generally, we denote by · Y the norm of a space Y . For simplicity, we write · L p ( ) by · p . If there is no ambiguity, we will omit the variables t and x. Let B 1 and B 2 be the constants with For a measurable function p : ⊂ R n → [1, ∞], the Lebesgue space is a Banach space equipped with Luxembourg-type norm for any y ∈ L p(·) ( ).

Blow-up results
In this section, we establish blow-up results of solutions with three levels of initial energy and estimate bounds of blow-up time. For this, we need ab auxiliary lemma.

Lemma 3.1 ([12]) Let B(t) be a positive, twice differentiable function verifying
Taking the scalar product (1.1) by w t in L 2 ( ) and using (1.3), we get d dt Using this and the relation we get and where

Case of non-positive initial energy
Let one of the following hold.

Lemma 3.2 Let w be the solution of problem
Then there exists a constant η * > η 1 such that It is easily seen that Since E(0) < E 1 , there exists η * > η 1 such that Since g is decreasing on (η 1 , ∞), we see that From (3.19), we also know We will show (3.20) by contradiction. Suppose that there exists t 0 ∈ [0, T * ) such that Because the solution w is continuous in t, there exists t 1 > 0 such that Noting that g is decreasing on (η 1 , ∞) and using (3.23), (3.25), (3.22), (3.24), (3.21), (3.1), we get This is a contradiction.