Blow-up of solutions for a Kirchhoff type equation with variable-exponent nonlinearities

Abstract

This paper deals with a Kirchhoff type equation with variable exponent nonlinearities, subject to a nonlinear boundary condition. Under appropriate conditions and regarding arbitrary positive initial energy, it is proved that solutions blow up in a finite time. Moreover, we obtain the upper bound estimate of the blow-up time.

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

“…Due to the great importance both theoretically and practically, numerous researchers have studied equations with nonstandard growth conditions, that is, equations with the variable exponent of nonlinearities, [31][32][33][34][35][36][37][38][39][40][41][42][43][44][45][46]. Compared with equations with variable exponent nonlinearities, less work has been done in a system of equations with variable exponent nonlinearities.…”

General decay and blow up of solutions for a class of inverse problem with elasticity term and variable‐exponent nonlinearities

“…Our approach in this section is based on concavity method. 6,14 In order to prove this result, we set a = 𝛽 = 0 and 𝜙(t) ≡ 1 and define v(x, t) = e −𝜆t u(x, t).…”

“…They proved a blowup in finite time with negative initial energy under suitable conditions on g, f and the variable exponent of the ⃗ p(x, t)-Laplace operator. For more results regarding this matter, we refer the reader to previous studies [13][14][15][16][17][18] and the review paper. 19 Motivated by the aforementioned works, in the present paper, we study a class of elastic inverse source problem with variable-exponent nonlinearities.…”

Blow up of solutions of Kirchhoff type viscoelastic wave equations with logarithmic nonlinearity of variable exponents

“…In [2], E.Pi.skin and Ayse Fidan proved the blow up result by a concavity with arbitrary positive initial energy. We refer the interested reader to see [3,4]. When G ∥∇u∥ 2 2 = 1, and the free term has no logarithmic, problem (1.1) become the following viscoelastic wave equation with variable exponents…”