scholarly journals A sufficient condition for blowup solutions of nonlinear heat equations

2004 ◽  
Vol 293 (1) ◽  
pp. 227-236 ◽  
Author(s):  
Shaohua Chen
Keyword(s):  
Heat Equations ◽  
Sufficient Condition ◽  
Nonlinear Heat Equations

scholarly journals Time analyticity of the biharmonic heat equation, the heat equation with potentials and some nonlinear heat equations

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Chulan Zeng
Keyword(s):  
Heat Equation ◽  
Growth Conditions ◽  
Nonlinear Heat Equation ◽  
Heat Equations ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Power Nonlinearity ◽  
Nonlinear Heat Equations ◽  
Ill Posed ◽  
Necessary And Sufficient

<p style='text-indent:20px;'>In this paper, we investigate the pointwise time analyticity of three differential equations. They are the biharmonic heat equation, the heat equation with potentials and some nonlinear heat equations with power nonlinearity of order <inline-formula><tex-math id="M1">\begin{document}$ p $\end{document}</tex-math></inline-formula>. The potentials include all the nonnegative ones. For the first two equations, we prove if <inline-formula><tex-math id="M2">\begin{document}$ u $\end{document}</tex-math></inline-formula> satisfies some growth conditions in <inline-formula><tex-math id="M3">\begin{document}$ (x,t)\in \mathrm{M}\times [0,1] $\end{document}</tex-math></inline-formula>, then <inline-formula><tex-math id="M4">\begin{document}$ u $\end{document}</tex-math></inline-formula> is analytic in time <inline-formula><tex-math id="M5">\begin{document}$ (0,1] $\end{document}</tex-math></inline-formula>. Here <inline-formula><tex-math id="M6">\begin{document}$ \mathrm{M} $\end{document}</tex-math></inline-formula> is <inline-formula><tex-math id="M7">\begin{document}$ R^d $\end{document}</tex-math></inline-formula> or a complete noncompact manifold with Ricci curvature bounded from below by a constant. Then we obtain a necessary and sufficient condition such that <inline-formula><tex-math id="M8">\begin{document}$ u(x,t) $\end{document}</tex-math></inline-formula> is analytic in time at <inline-formula><tex-math id="M9">\begin{document}$ t = 0 $\end{document}</tex-math></inline-formula>. Applying this method, we also obtain a necessary and sufficient condition for the solvability of the backward equations, which is ill-posed in general.</p><p style='text-indent:20px;'>For the nonlinear heat equation with power nonlinearity of order <inline-formula><tex-math id="M10">\begin{document}$ p $\end{document}</tex-math></inline-formula>, we prove that a solution is analytic in time <inline-formula><tex-math id="M11">\begin{document}$ t\in (0,1] $\end{document}</tex-math></inline-formula> if it is bounded in <inline-formula><tex-math id="M12">\begin{document}$ \mathrm{M}\times[0,1] $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M13">\begin{document}$ p $\end{document}</tex-math></inline-formula> is a positive integer. In addition, we investigate the case when <inline-formula><tex-math id="M14">\begin{document}$ p $\end{document}</tex-math></inline-formula> is a rational number with a stronger assumption <inline-formula><tex-math id="M15">\begin{document}$ 0&lt;C_3 \leq |u(x,t)| \leq C_4 $\end{document}</tex-math></inline-formula>. It is also shown that a solution may not be analytic in time if it is allowed to be <inline-formula><tex-math id="M16">\begin{document}$ 0 $\end{document}</tex-math></inline-formula>. As a lemma, we obtain an estimate of <inline-formula><tex-math id="M17">\begin{document}$ \partial_t^k \Gamma(x,t;y) $\end{document}</tex-math></inline-formula> where <inline-formula><tex-math id="M18">\begin{document}$ \Gamma(x,t;y) $\end{document}</tex-math></inline-formula> is the heat kernel on a manifold, with an explicit estimation of the coefficients.</p><p style='text-indent:20px;'>An interesting point is that a solution may be analytic in time even if it is not smooth in the space variable <inline-formula><tex-math id="M19">\begin{document}$ x $\end{document}</tex-math></inline-formula>, implying that the analyticity of space and time can be independent. Besides, for general manifolds, space analyticity may not hold since it requires certain bounds on curvature and its derivatives.</p>

scholarly journals On linear and nonlinear heat equations in degenerating domains

2017 ◽  
Author(s):  
M. Jenaliyev ◽  
M. Ramazanov ◽  
M. Yergaliyev
Keyword(s):  
Heat Equations ◽  
Nonlinear Heat Equations ◽  
Linear And Nonlinear

scholarly journals Blow-up of solutions of nonlinear heat equations

1988 ◽  
Vol 129 (2) ◽  
pp. 409-419 ◽  
Author(s):  
Luis A. Caffarrelli ◽  
Avner Friedman
Keyword(s):  
Blow Up ◽  
Heat Equations ◽  
Nonlinear Heat Equations

scholarly journals A blow-up dichotomy for semilinear fractional heat equations

2020 ◽  
Author(s):  
Robert Laister ◽  
Mikołaj Sierżęga
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Positive Solutions ◽  
Finite Time ◽  
Blow Up ◽  
Heat Equations ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Whole Space ◽  
Necessary And Sufficient

Abstract We derive a blow-up dichotomy for positive solutions of fractional semilinear heat equations on the whole space. That is, within a certain class of convex source terms, we establish a necessary and sufficient condition on the source for all positive solutions to become unbounded in finite time. Moreover, we show that this condition is equivalent to blow-up of all positive solutions of a closely-related scalar ordinary differential equation.

Sharp convolution and multiplication estimates in weighted spaces

2015 ◽  
Vol 13 (05) ◽  
pp. 457-480 ◽  
Author(s):  
Joachim Toft ◽  
Karoline Johansson ◽  
Stevan Pilipović ◽  
Nenad Teofanov
Keyword(s):  
Differential Equations ◽  
Stokes Equations ◽  
Weighted Spaces ◽  
Modulation Spaces ◽  
Local Analysis ◽  
Navier Stokes ◽  
Heat Equations ◽  
Navier Stokes Equations ◽  
Nonlinear Heat Equations ◽  
Hyperbolic Differential Equations

We establish sharp convolution and multiplication estimates in weighted Lebesgue, Fourier Lebesgue and modulation spaces. We cover, especially some results in [L. Hörmander, Lectures on Nonlinear Hyperbolic Differential Equations (Springer, Berlin, 1997); S. Pilipović, N. Teofanov and J. Toft, Micro-local analysis in Fourier Lebesgue and modulation spaces, II, J. Pseudo-Differ. Oper. Appl.1 (2010) 341–376]. The results are also related to some results by Iwabuchi in [T. Iwabuchi, Navier–Stokes equations and nonlinear heat equations in modulation spaces with negative derivative indices, J. Differential Equations248 (2010) 1972–2002].

The Flux Model of the Movement of Tumor Cells and Healthy Cells Using a System of Nonlinear Heat Equations

2011 ◽  
Vol 18 (12) ◽  
pp. 1831-1839 ◽  
Author(s):  
Meng-Rong Li ◽  
Yu-Ju Lin ◽  
Tzong-Hann Shieh
Keyword(s):  
Tumor Cells ◽  
Heat Equations ◽  
Nonlinear Heat Equations ◽  
Flux Model

ASYMPTOTICS FOR FRACTIONAL NONLINEAR HEAT EQUATIONS

2005 ◽  
Vol 72 (03) ◽  
pp. 663-688 ◽  
Author(s):  
NAKAO HAYASHI ◽  
ELENA I. KAIKINA ◽  
PAVEL I. NAUMKIN
Keyword(s):  
Heat Equations ◽  
Nonlinear Heat Equations
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