Global existence and blow-up of solutions to a semilinear heat equation with singular potential and logarithmic nonlinearity

Deng, Xiumei; Zhou, Jun

doi:10.3934/cpaa.2020042

Cited by 10 publications

(7 citation statements)

References 31 publications

(48 reference statements)

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“…With much literature concerning the polynomial nonlinearity, logarithmic nonlinearity also appears frequently in PDEs, see for example, [22][23][24][25][26][27][28][29][30][31][32] . In fact, this type nonlinearity can be applied in many branches of physics, such as optics 33 , inflationary cosmology 34,35 , geophysics 36,37 , and nuclear physics 38 .…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…Part 1: Behavior of weak solutions with (𝑢 0 , 𝑢 1 ) ∈ 𝔄 𝛿 . Let 𝑢 = 𝑢(𝑡), 𝑡 ∈ [0, 𝑇) be a weak solution of problem ( 1) with (𝑢 0 , 𝑢 1 ) ∈ 𝔄 𝛿 , where 𝔄 𝛿 is the set defined in (25) and 𝑇 is the maximum existence time. Then by Lemma 3, we have 𝐼(𝑢(𝑡)) < −𝛿 for all 𝑡 ∈ [0, 𝑇) and…”

Section: Proofs Of the Main Resultsmentioning

confidence: 99%

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Infinite time blow‐up of solutions to a class of wave equations with weak and strong damping terms and logarithmic nonlinearity

2021

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This paper investigates the infinite time blow‐up of solutions with arbitrary high initial energy to wave equations with weak damping term, strong damping term, and logarithmic nonlinearity. This problem has been studied previously with the assumptions that there is no strong damping term and the initial displacement and initial velocity have the same sign. However, from the physical point of view, it is obvious that the initial displacement and initial velocity may have different signs, and it is very necessary to consider the effects of the strong damping term. For example, the strong damping term indicates that the stress is not only proportional to the strain as with the Hooke law, but also proportional to the strain rate as in a linearized Kelvin–Voigt material. In this paper, by providing a completely different method from previous studies, we show that the solutions may blow up at infinity with arbitrary high initial energy when the model involves the strong damping term and the initial displacement and initial velocity may have different signs. Moreover, in this paper, we prove for the first time how to extend the solution over time (the whole half line) in studying the infinite time blow‐up phenomena for hyperbolic equations with logarithmic nonlinearity. These results fill in the gaps in previous studies on this type of models.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Section: Proofs Of the Main Resultsmentioning

confidence: 99%

Infinite time blow‐up of solutions to a class of wave equations with weak and strong damping terms and logarithmic nonlinearity

2021

Self Cite

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“…Since the first work done in 1979 by Galaktionov, Kurdyumov, Mikhailnov, and Samarskii [17], problems with logarithmic nonlinearities have received considerable attention in various research papers due to their multiple applications in physics and applied sciences. Readers can see the following works and the references [18], [19], [20], [21], [9], [10], [11], [14], [33], [34].…”

Section: Theorem 11 ([15]mentioning

confidence: 99%

Existence and non-existence of global solutions for a heat equation with degenerate coefficients

Castillo¹,

Guzmán-Rea²,

Zegarra³

2022

Preprint

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In this paper, we will study the following parabolic problem ut − div(ω(x)∇u) = h(t)f (u) + l(t)g(u) with non-negative initial conditions pertaining to C b (R N ), where the weight ω is an appropriate function that belongs to the Munckenhoupt class A 1+ 2 N and the functions f , g, h and l are non-negative and continuous. The main goal is to establish of global and non-global existence of non-negative solutions. In addition, to present the particular case when h(t) ∼ t r (r > −1), l(t) ∼ t s (s > −1), f (u) = u p and g(u) = (1 + u)[ln(1 + u)] p , we obtain both the so-called Fujita's exponent and the second critical exponent in the sense of Lee and Ni [32]. Our results extend those obtained by Fujishima et al. [15] who worked when h(t) = 1, l(t) = 0 and f (u) = u p .

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“…In the past years, many researchers have paid attention to the above problem (4) (see [2][3][4][5][6][7][8][9]). When the source f (u) is a polynomial nonlinearity, Tan [2] investigated the following non-Newton filtration equation with special medium…”

Section: Introductionmentioning

confidence: 99%

A singular non-Newton filtration equation with logarithmic nonlinearity: global existence and blow-up

Deng¹,

Zeng²,

Jiang³

2022

Comptes Rendus. Mécanique

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In this paper, we study the initial-boundary value problem of the singular non-Newton filtration equation with logarithmic nonlinearity. By using the concavity method, we obtain the existence of finite time blow-up solutions at initial energy J (u 0 ) d . Furthermore, we discuss the asymptotic behavior of the weak solution and prove that the weak solution converges to the corresponding stationary solution as t → +∞. Finally, we give sufficient conditions for global existence and blow-up of solutions at initial energy J (u 0 ) > d .

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Global existence and blow-up of solutions to a semilinear heat equation with singular potential and logarithmic nonlinearity

Cited by 10 publications

References 31 publications

Infinite time blow‐up of solutions to a class of wave equations with weak and strong damping terms and logarithmic nonlinearity

Infinite time blow‐up of solutions to a class of wave equations with weak and strong damping terms and logarithmic nonlinearity

Existence and non-existence of global solutions for a heat equation with degenerate coefficients

A singular non-Newton filtration equation with logarithmic nonlinearity: global existence and blow-up

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