Classification of certain qualitative properties of solutions for the quasilinear parabolic equations

Li, Yan; Zhang, Zhengce; Zhu, Liu

doi:10.1007/s11425-016-9077-8

Cited by 16 publications

(8 citation statements)

References 30 publications

(52 reference statements)

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“…Remark 5.3. Note that in the limiting case α → 1 and s → 1, the results of Theorem 5.2 coincides with the results obtained in [25].…”

Section: Global Solutionsupporting

confidence: 86%

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Qualitative properties of solutions to the nonlinear time-space fractional diffusion equation

B.¹,

Ruzhansky²,

Torebek³

2022

Preprint

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In the present paper, we study the Cauchy-Dirichlet problem to the nonlocal nonlinear diffusion equation with polynomial nonlinearities0|t and space-fractional p-Laplacian operator (−∆) s p . We give a simple proof of the comparison principle for the considered problem using purely algebraic relations, for different sets of γ, µ, m and q. The Galerkin approximation method is used to prove the existence of a local weak solution. The blow-up phenomena, existence of global weak solutions and asymptotic behavior of global solutions are classified using the comparison principle.

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“…Remark 5.3. Note that in the limiting case α → 1 and s → 1, the results of Theorem 5.2 coincides with the results obtained in [25].…”

Section: Global Solutionsupporting

confidence: 86%

“…In the case α = s = 1, γ = −1, the problem (1.1) coincides with a quasilinear parabolic equation which has been studied by Li et al in [25]    u t − div(|∇| p−2 ∇u) = −|u| m−1 u + µ|u| q−2 u, x ∈ Ω, t > 0, u(x, t) = 0, x ∈ ∂Ω, t > 0, u(x, 0) = u 0 (x), x ∈ Ω, (…”

Section: Introductionmentioning

confidence: 98%

Qualitative properties of solutions to the nonlinear time-space fractional diffusion equation

B.¹,

Ruzhansky²,

Torebek³

2022

Preprint

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“…For a comparison principle for weak solutions of p-Laplacian heat equation in a bounded domain, we refer to recent works [LZZ18] when G = (R n , +), and to [RS18] when G is a graded Lie group, the latter also allowing more general hypoelliptic differential operators (Rockland operators).…”

Section: Introduction and Resultsmentioning

confidence: 99%

Existence and non-existence of global solutions for semilinear heat equations and inequalities on sub-Riemannian manifolds, and Fujita exponent on unimodular Lie groups

Ruzhansky,

Yessirkegenov

2018

Preprint

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In this paper we study the global well-posedness of the following Cauchy problem on a sub-Riemannian manifold M :for u 0 ≥ 0, where L M is a sub-Laplacian of M . In the case when M is a connected unimodular Lie group G, which has polynomial volume growth, we obtain a critical Fujita exponent, namely, we prove that all solutions of the Cauchy problem with u 0 ≡ 0, blow up in finite time if and only if 1 < p ≤ p F := 1 + 2/D when f (u) ≃ u p , where D is the global dimension of G. In the case 1 < p < p F and when f : [0, ∞) → [0, ∞) is a locally integrable function such that f (u) ≥ K 2 u p for some K 2 > 0, we also show that the differential inequalitydoes not admit any nontrivial distributional (a function u ∈ L p loc (Q) which satisfies the differential inequality in D ′ (Q)) solution u ≥ 0 in Q := (0, ∞)×G. Furthermore, in the case when G has exponential volume growth and f : [0, ∞) → [0, ∞) is a continuous increasing function such that f (u) ≤ K 1 u p for some K 1 > 0, we prove that the Cauchy problem has a global, classical solution for 1 + 2/d < p < ∞ and some positive u 0 ∈ L q (G) with 1 ≤ q < ∞, where d is the local dimension of G. Moreover, we also discuss all these results in more general settings of sub-Riemannian manifolds M .

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“…The goal of this note is to give a simple proof of a comparison principle for the initial boundary value problem for nonlinear heat Rockland operators on graded groups using pure algebraic relations, inspired by the recent work . For thorough analysis of sub‐harmonic analysis in related settings see, for example, , see also, for example, [, ] for some related analysis and references therein.…”

Section: Introductionmentioning

confidence: 99%

A comparison principle for nonlinear heat Rockland operators on graded groups

Ruzhansky

Сураган

2018

Bull. London Math. Soc.

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Classification of certain qualitative properties of solutions for the quasilinear parabolic equations

Cited by 16 publications

References 30 publications

Qualitative properties of solutions to the nonlinear time-space fractional diffusion equation

Qualitative properties of solutions to the nonlinear time-space fractional diffusion equation

Existence and non-existence of global solutions for semilinear heat equations and inequalities on sub-Riemannian manifolds, and Fujita exponent on unimodular Lie groups

A comparison principle for nonlinear heat Rockland operators on graded groups

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