Blowing up at zero points of potential for an initial boundary value problem

Guo, Jong‐Shenq; Shimojō, Masahiko

doi:10.3934/cpaa.2011.10.161

Cited by 22 publications

(18 citation statements)

References 3 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Then there exist symmetric solutions such that 0 is a blowup point [16]; the latter condition is only known to hold when Ω is convex and the potential V (x) is monotonically decreasing near the boundary.…”

Section: Remark 12 Let Us Summarize the Previously Known Results Abmentioning

confidence: 99%

“…Let us now explain the difficulties in ruling out the possibility of blowup at zero points of the potential V , and the new ideas to overcome these difficulties. It is already known [16,15] The outline of the paper is as follows. In Section 2, we prove a nondegeneracy result which is one of the ingredients of the proof of Theorem 1.2.…”

Section: Remark 12 Let Us Summarize the Previously Known Results Abmentioning

confidence: 99%

“…This was proved in [16]. For completeness, we reproduce the (supersolution) argument of [16] as follows.…”

Section: )mentioning

confidence: 92%

“…Ω is a domain of R n , ( Our first main result rules out the possibility of blowup at zero points of the potential V for monotone in time solutions of (1.1), under suitable assumptions. In fact, the case of the homogeneous Dirichlet problem associated with equation (1.1) with f (0) = 0 was completely solved in [16]. The more delicate case f (0) > 0 was left as an open problem.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Excluding blowup at zero points of the potential by means of Liouville-type theorems

Guo

Souplet

2018

Journal of Differential Equations

Self Cite

View full text Add to dashboard Cite

We prove a local version of a (global) result of Merle and Zaag about ODE behavior of solutions near blowup points for subcritical nonlinear heat equations. As an application, for the equation ut = ∆u + V (x)f (u), we rule out the possibility of blowup at zero points of the potential V for monotone in time solutions when f (u) ∼ u p for large u, both in the Sobolev subcritical case and in the radial case. This solves a problem left open in previous work on the subject. Suitable Liouville-type theorems play a crucial role in the proofs.

show abstract

Section: Remark 12 Let Us Summarize the Previously Known Results Abmentioning

confidence: 99%

Section: Remark 12 Let Us Summarize the Previously Known Results Abmentioning

confidence: 99%

“…This was proved in [16]. For completeness, we reproduce the (supersolution) argument of [16] as follows.…”

Section: )mentioning

confidence: 92%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Excluding blowup at zero points of the potential by means of Liouville-type theorems

Guo

Souplet

2018

Journal of Differential Equations

Self Cite

View full text Add to dashboard Cite

show abstract

“…Several conditions which ensure non-blow-up at the zero point were obtained in [15,16,18]. Examples of solutions blowing up at x = 0, in contrast, were found in [17,11]. Filippas and Tertikas [11] constructed self-similar solutions that blow up (in finite time) at x = 0 in the cases p < p S (2a) or p S (2a) < p < p JL (2a), where…”

mentioning

confidence: 99%

Refined construction of type II blow-up solutions for semilinear heat equations with Joseph–Lundgren supercritical nonlinearity

Mukai¹,

Seki²

2021

DCDS

View full text Add to dashboard Cite

We are concerned with blow-up mechanisms in a semilinear heat equation:ut = ∆u + |x| 2a u p , x ∈ R N , t > 0, where p > 1 and a > −1 are constants. As for the Fujita equation, which corresponds to a = 0, a well-known result due to M. A. Herrero and J. J. L. Velázquez, C. R. Acad. Sci. Paris Sér. I Math. (1994), states that if N ≥ 11 andwhere T is the blow-up time. We revisit the idea of their construction and obtain refined estimates for such solutions by the techniques developed in recent works and elaborate estimates of the heat semigroup in backward similarity variables. Our method is naturally extended to the case a = 0. As a consequence, we obtain an example of solutions that blow up at x = 0, the zero point of potential |x| 2a with a > 0, and behave in non-self-similar manner for N > 10 + 8a. This last result is in contrast to backward self-similar solutions previously obtained for N < 10 + 8a, which blow up at x = 0.

show abstract

Blowup at space infinity for solutions of a system of nonautonomous semilinear heat equations

Cabral-García

Villa-Morales

2019

Math Methods in App Sciences

View full text Add to dashboard Cite

In this paper, we will see that the global or local existence of solutions to depends on the initial datums and the global or local existence of solutions to We also give some bounds for the maximal existence time of the partial differential system. Moreover, if such existence time is finite and , then we will prove the partial differential system has solutions that blows up at space infinite.

show abstract

Blowing up at zero points of potential for an initial boundary value problem

Cited by 22 publications

References 3 publications

Excluding blowup at zero points of the potential by means of Liouville-type theorems

Excluding blowup at zero points of the potential by means of Liouville-type theorems

Refined construction of type II blow-up solutions for semilinear heat equations with Joseph–Lundgren supercritical nonlinearity

Blowup at space infinity for solutions of a system of nonautonomous semilinear heat equations

Contact Info

Product

Resources

About