Convergence to travelling fronts in semilinear parabolic equations

1978 ◽  
Vol 80 (3-4) ◽  
pp. 213-234 ◽  
Author(s):  
Franz Rothe
Keyword(s):  
Asymptotic Behavior ◽  
Initial Data ◽  
Parabolic Equations ◽  
Step Function ◽  
Wave Type ◽  
All Solutions ◽  
Asymptotic Speed ◽  
Rate Of Decay ◽  
Travelling Fronts ◽  
Semilinear Parabolic

SynopsisWe study the convergence to the stationary state for the parabolic equation u, = uxx + F(u). There exist wave-type solutions u(x, t) = φ(x − ct) for a continuum of velocities c. In the asymptotic behavior of this equation was investigated for a step function as initial data. In this paper we obtain the asymptotic behavior for a large class of monotone initial data.All solutions with initial data in this class evolve to wave-type solutions, where the rate of decay of the initial data determines the asymptotic speed.

scholarly journals Solvability for time-fractional semilinear parabolic equations with singular initial data

2021 ◽  
Author(s):  
Marius Ghergu ◽  
Yasuhito Miyamoto ◽  
Masamitsu Suzuki
Keyword(s):  
Initial Data ◽  
Parabolic Equations ◽  
Critical Case ◽  
Semilinear Parabolic Equations ◽  
Time Solution ◽  
Existence And Nonexistence ◽  
Singular Initial Data ◽  
Semilinear Parabolic

We discuss the existence and nonexistence of a local and global-in-time solution to the fractional problem $$ ¥begin{cases} ¥partial_t^{¥alpha}u=¥Delta u+f(u) & x¥in¥Omega,¥ 01$ one has $|f(¥xi)-f(¥eta)|¥le C(1+|¥xi|+|¥eta|)^{p-1}|¥xi-¥eta|$ for all $¥xi, ¥eta¥in ¥R$. Particular attention is paid to the doubly critical case $(p,r)=(1+2/N,1)$.

scholarly journals Existence and asymptotic behavior of Radon measure-valued solutions for a class of nonlinear parabolic equations

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Quincy Stévène Nkombo ◽  
Fengquan Li ◽  
Christian Tathy
Keyword(s):  
Asymptotic Behavior ◽  
Initial Data ◽  
Parabolic Equations ◽  
Radon Measure ◽  
Decay Estimate ◽  
Young Measure ◽  
Nonlinear Parabolic Equations ◽  
Bounded Interval ◽  
Nonlinear Parabolic ◽  
Alpha Beta

AbstractIn this paper we address the weak Radon measure-valued solutions associated with the Young measure for a class of nonlinear parabolic equations with initial data as a bounded Radon measure. This problem is described as follows: $$ \textstyle\begin{cases} u_{t}=\alpha u_{xx}+\beta [\varphi (u) ]_{xx}+f(u) &\text{in} \ Q:=\Omega \times (0,T), \\ u=0 &\text{on} \ \partial \Omega \times (0,T), \\ u(x,0)=u_{0}(x) &\text{in} \ \Omega , \end{cases} $$ { u t = α u x x + β [ φ ( u ) ] x x + f ( u ) in Q : = Ω × ( 0 , T ) , u = 0 on ∂ Ω × ( 0 , T ) , u ( x , 0 ) = u 0 ( x ) in Ω , where $T>0$ T > 0 , $\Omega \subset \mathbb{R}$ Ω ⊂ R is a bounded interval, $u_{0}$ u 0 is nonnegative bounded Radon measure on Ω, and $\alpha , \beta \geq 0$ α , β ≥ 0 , under suitable assumptions on φ and f. In this work we prove the existence and the decay estimate of suitably defined Radon measure-valued solutions for the problem mentioned above. In particular, we study the asymptotic behavior of these Radon measure-valued solutions.

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