On the critical exponent for some semilinear reaction–diffusion systems on general domains

Castillo, Ricardo Moreno; Loayza, Miguel

doi:10.1016/j.jmaa.2015.03.035

Cited by 23 publications

(13 citation statements)

References 14 publications

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“…For the non-global existence results in Theorem 1.1, the authors powerfully used Lemma 3.1 in [15], which was obtained through an iterative method, that has been widely used in several other works, e.g. see [2], [12], [5], [6], [7]. Unfortunately, this same approach cannot be used for the case of problem (3), due to logarithmic nonlinearity (u + 1)(ln(u + 1)) q .…”

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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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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“…However, for a general source term ( ) , there is no paper which discuss the necessary and sufficient condition for the existence and nonexistence of the global solutions. Because of this fact, recent researches for the existence and nonexistence of global solutions have been studied based on Meier's criterion which were not the necessary and sufficient condition (for example, see 4,5 ).…”

Section: Introductionmentioning

confidence: 99%

“…( ) = ( ) ( ) for > 0 and > 0. In general, multiplicative property of the source term is strongly used to obtain the existence and nonexistence of global solutions in the parabolic equations (for example, see 2,4,14 ). For a general source term ( ) ( ), there is no paper on necessary and sufficient condition for the existence of global solution.…”

Section: Introductionmentioning

confidence: 99%

A New Necessary and Sufficient Condition for the Existence of Global Solutions to Semilinear Parabolic Equations on bounded domains

Chung

Hwang

2022

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The purpose of this paper is to give a necessary and sufficient condition for the existence and non-existence of global solutions of the following semilinear parabolic equations \[ u_{t}=\Delta u+\psi(t)f(u),\,\,\mbox{ in }\Omega\times (0,t^{*}), \] under the Dirichlet boundary condition on a bounded domain. In fact, this has remained as an open problem for a few decades, even for the case $f(u)=u^{p}$. As a matter of fact, we prove:\\ \[ \begin{aligned} &\mbox{there is no global solution for any initial data if and only if }\\ &\mbox{the function } f \mbox{ satisfies}\\ &\hspace{20mm}\int_{0}^{\infty}\psi(t)\frac{f\left(\epsilon \,\left\| S(t)u_{0}\right\|_{\infty}\right)}{\left\| S(t)u_{0}\right\|_{\infty}}dt=\infty\\ &\mbox{for every }\,\epsilon>0\,\mbox{ and nonnegative nontrivial initial data }\,u_{0}\in C_{0}(\Omega). \end{aligned} \] Here, $(S(t))_{t\geq 0}$ is the heat semigroup with the Dirichlet boundary condition.

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“…However, the necessary and sufficient condition for the general source term ψ(t)u p has remained as an open problem for a few decades. The open problem has faced methodological limitations and recent researches have adopted Meier's criterion which were not the necessary and sufficient condition (for example, see [4,5]). In conclusion, there has been no progress in research on necessary and sufficient conditions for the general source term ψ(t)u p as well as ψ(t)f (u).…”

Section: Introductionmentioning

confidence: 99%

A New Necessary and Sufficient Condition for the Existence of Global Solutions to Semilinear Parabolic Equations on Bounded Domains

Chung¹,

Hwang²

2022

Preprint

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The purpose of this paper is to give a necessary and sufficient condition for the existence and non-existence of global solutions of the following semilinear parabolic equationsunder the Dirichlet boundary condition on a bounded domain. In fact, this has remained as an open problem for a few decades, even for the case f (u) = u p . As a matter of fact, we prove: there is no global solution for any initial data if and only if the function f satisfiesfor every ǫ > 0 and nonnegative nontrivial initial data u 0 ∈ C 0 (Ω).Here, (S(t)) t≥0 is the heat semigroup with the Dirichlet boundary condition.

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On the critical exponent for some semilinear reaction–diffusion systems on general domains

Cited by 23 publications

References 14 publications

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

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

A New Necessary and Sufficient Condition for the Existence of Global Solutions to Semilinear Parabolic Equations on bounded domains

A New Necessary and Sufficient Condition for the Existence of Global Solutions to Semilinear Parabolic Equations on Bounded Domains

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