Cauchy problems of pseudo-parabolic equations with inhomogeneous terms

Li, Zhongping; Du, Wanjuan

doi:10.1007/s00033-015-0558-2

Cited by 14 publications

(12 citation statements)

References 38 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…(2) If we denote by Φ the set of weak solutions to (2), then by the definitions of J in (12) and N in (15), we have…”

Section: Notations and Main Resultsmentioning

confidence: 99%

“…(1) If J(φ) < 0, then we can easily get from the definitions of J and I in (12) and (13) respectively that I(φ) < 0. So we have φ ∈ W if J(φ) < 0.…”

Section: Notations and Main Resultsmentioning

confidence: 99%

“…Li and Du [12] studied the Cauchy problem of equation in (1) with σ > 0. They got the critical Fujita exponent (p c ) and second critical exponent (α c ) by the integral representation and comparison principle.…”

Section: Introductionmentioning

confidence: 99%

“…They got the critical Fujita exponent (p c ) and second critical exponent (α c ) by the integral representation and comparison principle. The main results obtained in [12] are as follows:…”

Section: Introductionmentioning

confidence: 99%

“…problem (1), and the parameters satisfies (3), which allows σ to be less than 0. To overcome the singularity of |x| σ at x = 0, we use potential well method by introducing the |x| σ weighed-L p+1 (Ω) space and assume there is a lower bound of σ, i.e, The main results of this paper can be summarized as follows: Let J and I be the functionals given in (12) and (13), respectively; d be the mountain-pass level given in (14); S ρ and S ρ be the sets defined in (20).…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Initial boundary value problem for a inhomogeneous pseudo-parabolic equation

Zhou

2020

era

View full text Add to dashboard Cite

This paper deals with the global existence and blow-up of solutions to a inhomogeneous pseudo-parabolic equation with initial value u 0 in the Sobolev space H 1 0 (Ω), where Ω ⊂ R n (n ≥ 1 is an integer) is a bounded domain. By using the mountain-pass level d (see (14)), the energy functional J (see (12)) and Nehari function I (see (13)), we decompose the space H 1 0 (Ω) into five parts, and in each part, we show the solutions exist globally or blow up in finite time. Furthermore, we study the decay rates for the global solutions and lifespan (i.e., the upper bound of blow-up time) of the blow-up solutions. Moreover, we give a blow-up result which does not depend on d. By using this theorem, we prove the solution can blow up at arbitrary energy level, i.e. for any M ∈ R, there exists u 0 ∈ H 1 0 (Ω) satisfying J(u 0) = M such that the corresponding solution blows up in finite time.

show abstract

“…(2) If we denote by Φ the set of weak solutions to (2), then by the definitions of J in (12) and N in (15), we have…”

Section: Notations and Main Resultsmentioning

confidence: 99%

“…(1) If J(φ) < 0, then we can easily get from the definitions of J and I in (12) and (13) respectively that I(φ) < 0. So we have φ ∈ W if J(φ) < 0.…”

Section: Notations and Main Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Initial boundary value problem for a inhomogeneous pseudo-parabolic equation

Zhou

2020

era

View full text Add to dashboard Cite

show abstract

Variations of heat equation on the half‐line via the Fokas method

Chatziafratis,

Fokas,

Aifantis

2024

Math Methods in App Sciences

View full text Add to dashboard Cite

In this review paper, we discuss some of our recent results concerning the rigorous analysis of initial boundary value problems (IBVPs) and newly discovered effects for certain evolution partial differential equations (PDEs). These equations arise in the applied sciences as models of phenomena and processes pertaining, for example, to continuum mechanics, heat‐mass transfer, solid–fluid dynamics, electron physics and radiation, chemical and petroleum engineering, and nanotechnology. The mathematical problems we address include certain well‐known classical variations of the traditional heat (diffusion) equation, including (i) the Sobolev–Barenblatt pseudoparabolic PDE (or modified heat or second‐order fluid equation), (ii) a fourth‐order heat equation and the associated Cahn–Hilliard (or Kuramoto–Sivashinsky) model, and (iii) the Rubinshtein–Aifantis double‐diffusion system. Our work is based on the synergy of (i) the celebrated Fokas unified transform method (UTM) and (ii) a new approach to the rigorous analysis of this method recently introduced by one of the authors. In recent works, we considered forced versions of the aforementioned PDEs posed in a spatiotemporal quarter‐plane with arbitrary, fully non‐homogeneous initial and boundary data, and we derived formally effective solution representations, for the first time in the history of the models, justifying a posteriori their validity. This included the reconstruction of the prescribed initial and boundary conditions, which required careful analysis of the various integral terms appearing in the formulae, proving that they converge in a strictly defined sense. In each IBVP, the novel formula was utilized to rigorously deduce the solution's regularity properties near the boundaries of the spatiotemporal domain. Importantly, this analysis is indispensable for proving (non)uniqueness of solution. These works extend previous investigations. The usefulness of our closed‐form solutions will be demonstrated by studying their long‐time asymptotics. Specifically, we will briefly review some asymptotic results about Barenblatt's equation.

show abstract

On a class of nonlinear time‐fractional pseudo‐parabolic equations with bounded delay

Wang

Pang

Zhou

et al. 2023

Math Methods in App Sciences

View full text Add to dashboard Cite

In the present paper, we study a class of nonlinear time‐fractional pseudo‐parabolic equations with bounded time‐delay. Firstly, we analyze the well‐posedness of the problem by the well‐known Galerkin's method and some compactness criteria. Next, the existence of the stationary solutions is established by utilizing a lemma of ours, which is a corollary of the Brouwer fixed point theorem. Finally, with regard to the stability analysis of the stationary solutions, we prove the local stability via the direct method of energy estimations and show the asymptotic stability by the construction of appropriate Lyapunov functionals.

show abstract

Cauchy problems of pseudo-parabolic equations with inhomogeneous terms

Cited by 14 publications

References 38 publications

Initial boundary value problem for a inhomogeneous pseudo-parabolic equation

Initial boundary value problem for a inhomogeneous pseudo-parabolic equation

Variations of heat equation on the half‐line via the Fokas method

On a class of nonlinear time‐fractional pseudo‐parabolic equations with bounded delay

Contact Info

Product

Resources

About