Abstract
This paper is devoted to the initial and boundary value problems for a class of nonlinear metaparabolic equationsut−βuxx−kuxxt+γuxxxx=fuxx. At low initial energy level (Ju0<d), we not only prove the existence of global weak solutions for these problems by the combination of the Galerkin approximation and potential well methods but also obtain the finite time blow-up result by adopting the potential well and improved concavity skills. Finally, we also discussed the finite time blow-up phenomenon for certain solutions of these problems with high initial energy.
Highlights
In this paper, we study the initial and boundary value problems for the following nonlinear metaparabolic equations ut − βuxx − kuxxt + γuxxxx = f ðuxÞx, x ∈ Ω, t > 0, ð1Þ uðx, 0Þ = u0ðxÞ, x ∈ Ω, ð2Þ uð0, tÞ = uð1, tÞ = 0, uxxð0, tÞ = uxxð1, tÞ = 0, t ≥ 0, ð3Þ in a bounded domain Ω = ð0, 1Þ, where u0ðxÞ is the initial value function defined on Ω, k > 0 is the viscosity coefficient, γ > 0 is the interfacial energy parameter, and the nonlinear smooth function f ðsÞ satisfies the following assumptions: 8 >< ðiÞj f ðsÞj ≤ αjsjq, α > < q +∞, ∀s ∈
We mainly study the qualitative properties of the solutions for the initial and boundary value problems (1)–(3)
It is well known that Equation (1) is a typical higher-order metaparabolic equation, which has extensive practical background and rich theoretical connotation
Summary
For the fourth-order nonlinear parabolic and hyperbolic equations, there are some results about the initial boundary value and Cauchy problems, especially on global existence/nonexistence, uniqueness/nonuniqueness, and asymptotic behavior [18,19,20,21,22,23,24,25]. Yang [26] considered the initial and boundary value problems of the following equation utt + λut + αuxxxx = f ðuxÞx, x ∈ ð0, 1Þ, t > 0: ð11Þ He studied the asymptotic property of the solution and gave some sufficient conditions of the blow-up. Xu et al [28] considered the initial and boundary value problems and proved the global existence and nonexistence of solutions by adopting and modifying the so called concavity method under some conditions with low initial energy. We investigate the finite time blow-up for certain solutions of problems (1)–(3) with high initial energy
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