Infinite time blow-up for half-harmonic map flow from R into S1

2021 ◽  
Vol 143 (4) ◽  
pp. 1261-1335
Author(s):  
Yannick Sire ◽  
Juncheng Wei ◽  
Youquan Zheng
Keyword(s):  
Blow Up ◽  
Harmonic Map ◽  
Infinite Time ◽  
Harmonic Map Flow

scholarly journals Blow-up for the 3-dimensional axially symmetric harmonic map flow into \begin{document}$ S^2 $\end{document}

2019 ◽  
Vol 39 (12) ◽  
pp. 6913-6943
Author(s):  
Juan Dávila ◽  
◽  
Manuel Del Pino ◽  
Catalina Pesce ◽  
Juncheng Wei ◽  
...  
Keyword(s):  
Blow Up ◽  
Harmonic Map ◽  
Axially Symmetric ◽  
3 Dimensional ◽  
Harmonic Map Flow

scholarly journals Uniqueness and nonuniqueness of limits of Teichmüller harmonic map flow

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
James Kohout ◽  
Melanie Rupflin ◽  
Peter M. Topping
Keyword(s):  
Constant Curvature ◽  
Harmonic Map ◽  
Gradient Flow ◽  
Minimal Immersion ◽  
Curvature Surface ◽  
Previous Theory ◽  
Target Manifold ◽  
Harmonic Map Flow

AbstractThe harmonic map energy of a map from a closed, constant-curvature surface to a closed target manifold can be seen as a functional on the space of maps and domain metrics. We consider the gradient flow for this energy. In the absence of singularities, previous theory established that the flow converges to a branched minimal immersion, but only at a sequence of times converging to infinity, and only after pulling back by a sequence of diffeomorphisms. In this paper, we investigate whether it is necessary to pull back by these diffeomorphisms, and whether the convergence is uniform as {t\to\infty}.

scholarly journals Global existence, energy decay and blow-up of solutions for wave equations with time delay and logarithmic source

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Sun-Hye Park
Keyword(s):  
Time Delay ◽  
Global Existence ◽  
Wave Equations ◽  
Blow Up ◽  
Logarithmic Sobolev Inequality ◽  
Decay Rates ◽  
Infinite Time ◽  
Global Existence Of Solutions ◽  
Frictional Damping ◽  
T Tau

AbstractIn this paper, we study the wave equation with frictional damping, time delay in the velocity, and logarithmic source of the form $$ u_{tt}(x,t) - \Delta u (x,t) + \alpha u_{t} (x,t) + \beta u_{t} (x, t- \tau ) = u(x,t) \ln \bigl\vert u(x,t) \bigr\vert ^{\gamma } . $$ u t t ( x , t ) − Δ u ( x , t ) + α u t ( x , t ) + β u t ( x , t − τ ) = u ( x , t ) ln | u ( x , t ) | γ . There is much literature on wave equations with a polynomial nonlinear source, but not much on the equations with logarithmic source. We show the local and global existence of solutions using Faedo–Galerkin’s method and the logarithmic Sobolev inequality. And then we investigate the decay rates and infinite time blow-up for the solutions through the potential well and perturbed energy methods.

scholarly journals Transition of blow-up mechanisms in k-equivariant harmonic map heat flow

Nonlinearity ◽  
2020 ◽  
Vol 33 (6) ◽  
pp. 2756-2796
Author(s):  
Paweł Biernat ◽  
Yukihiro Seki
Keyword(s):  
Heat Flow ◽  
Blow Up ◽  
Harmonic Map ◽  
Harmonic Map Heat Flow

scholarly journals Moving mesh for the axisymmetric harmonic map flow

2005 ◽  
Vol 39 (4) ◽  
pp. 781-796
Author(s):  
Benoit Merlet ◽  
Morgan Pierre
Keyword(s):  
Harmonic Map ◽  
Moving Mesh ◽  
Harmonic Map Flow

scholarly journals Asymptotics of the Teichmüller harmonic map flow

2013 ◽  
Vol 244 ◽  
pp. 874-893 ◽  
Author(s):  
Melanie Rupflin ◽  
Peter M. Topping ◽  
Miaomiao Zhu
Keyword(s):  
Harmonic Map ◽  
Harmonic Map Flow

scholarly journals Blow up and globality of solutions for a nonautonomous semilinear heat equation with Dirichlet condition

2019 ◽  
Vol 53 (1) ◽  
pp. 57-72
Author(s):  
Marcos Josías Ceballos-Lira ◽  
Aroldo Pérez
Keyword(s):  
Global Existence ◽  
Heat Equation ◽  
Mild Solution ◽  
Blow Up ◽  
Local Existence ◽  
Dirichlet Condition ◽  
Infinite Time ◽  
Diffusion Operator ◽  
Intrinsic Ultracontractivity ◽  
Semilinear Heat Equation

In this paper we prove the local existence of a nonnegative mild solution for a nonautonomous semilinear heat equation with Dirichlet condition, and give sucient conditions for the globality and for the blow up infinite time of the mild solution. Our approach for the global existence goes back to the Weissler's technique and for the nite time blow up we uses the intrinsic ultracontractivity property of the semigroup generated by the diffusion operator.

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