Semilinear Heat Equation Research Articles

Refined behavior and structural universality of the blow-up profile for the semilinear heat equation with non scale invariant nonlinearity

Refined behavior and structural universality of the blow-up profile for the semilinear heat equation with non scale invariant nonlinearity

Blow-up of the critical norm for a supercritical semilinear heat equation

We consider the scaling critical Lebesgue norm of blow-up solutions to the semilinear heat equation u_{t}=\Delta u+|u|^{p-1}u in an arbitrary C^{2+\alpha} domain of \mathbf{R}^{n} . In the range p>p_{S}:=(n+2)/(n-2) , we show that the critical norm must be unbounded near the blow-up time, where the type I blow-up condition is not imposed. The range p>p_{S} is optimal in view of the existence of type II blow-up solutions with bounded critical norm for p=p_{S} .

The Fujita exponent for a semilinear heat equation with forcing term on Heisenberg group

The Fujita exponent for a semilinear heat equation with forcing term on Heisenberg group

Hunter Self-Similar Implosion Profiles for the Gravitational Euler–Poisson System

Our result is a construction of infinitely many radial self-similar implosion profiles for the gravitational Euler–Poisson system. The problem can be expressed as solving a system of non-autonomous non-linear ODEs. The first rigorous existence result for a non-trivial solution to these ODEs is due to Guo et al. (Commun Math Phys 386(3):1551–1601, 2021), in which they construct a solution found numerically by Larson (Mon Not R Astron Soc 145(3):271–295, 1969) and Penston (Mon Not R Astron Soc 144(4):425–448, 1969) independently. The solutions we construct belong to a different regime and correspond to a strict subset of the family of profiles discovered numerically by Hunter (Astrophys J 218:834, 1977). Our proof adapts a technique developed by Collot et al. (Mem Am Math Soc 260(1255):v+97, 2019), in which they study blowup for a family of energy-supercritical focusing semilinear heat equations. In our case, the quasilinearity presents complications, most severely near the sonic point where the system degenerates.

Approximate Controllability of Semilinear Heat Equation with Noninstantaneous Impulses, Memory, and Delay

Approximate Controllability of Semilinear Heat Equation with Noninstantaneous Impulses, Memory, and Delay

Results of existence and uniqueness for the Cauchy problem of semilinear heat equations on stratified Lie groups

The aim of this paper is to give existence and uniqueness results for solutions of the Cauchy problem for semilinear heat equations on stratified Lie groups G with the homogeneous dimension N. We consider the nonlinear function behaves like |u|α or |u|α−1u(α>1) and the initial data u0 belongs to the Sobolev spaces Lsp(G) for 1<p<∞ and 0<s<N/p. Since stratified Lie groups G include the Euclidean space Rn as an example, our results are an extension of the existence and uniqueness results obtained by F. Ribaud on Rn to G. It should be noted that our proof is very different from it given by Ribaud on Rn. We adopt the generalized fractional chain rule on G to obtain the estimate for the nonlinear term, which is very different from the paracomposition technique adopted by Ribaud on Rn. By using the generalized fractional chain rule on G, we can avoid the discussion of Fourier analysis on G and make the proof more simple.

On degenerate blow-up profiles for the subcritical semilinear heat equation

We consider the semilinear heat equation with a superlinear power nonlinearity in the Sobolev subcritical range. We construct a solution which blows up in finite time only at the origin, with a completely new blow-up profile, which is cross-shaped. Our method is general and extends to the construction of other solutions blowing up only at the origin, with a large variety of blow-up profiles, degenerate or not.

Blow-up phenomenon to the semilinear heat equation for unbounded Laplacians on graphs

Blow-up phenomenon to the semilinear heat equation for unbounded Laplacians on graphs

On directional blow-up for a semilinear heat equation with space-dependent reaction

We consider nonnegative solutions u of the Cauchy problem for a semilinear heat equation with space-dependent reaction: ut=Δu+μ(x)up, u(x,0)=u0(x), where μ(x)≥0 satisfies some condition and the initial data u0(x)(≢0) satisfies ‖μ˜u0‖L∞(RN)<∞ with μ˜=μ1/(p−1). We study weighted solutions μ˜u which blow up at minimal blow-up time. Such a weighted solution blows up at space infinity in some direction (directional blow-up). We call this direction a blow-up direction of μ˜u. We give a sufficient and necessary condition on u0 for a weighted solution to blow up at minimal blow-up time. Moreover, we completely characterize blow-up directions of μ˜u by the profile of the initial data.

Fujita exponent for the global-in-time solutions to a semilinear heat equation with non-homogeneous weights

Fujita exponent for the global-in-time solutions to a semilinear heat equation with non-homogeneous weights

Local solvability and dilation-critical singularities of supercritical fractional heat equations

We consider the Cauchy problem for fractional semilinear heat equations with supercritical nonlinearities and establish both necessary conditions and sufficient conditions for local-in-time solvability. We introduce the notion of a dilation-critical singularity (DCS) of the initial data and show that such singularities always exist for a large class of supercritical nonlinearities. Moreover, we provide exact formulae for such singularities.

Unconditional uniqueness and non-uniqueness for Hardy–Hénon parabolic equations

We study the problems of uniqueness for Hardy–Hénon parabolic equations, which are semilinear heat equations with the singular potential (Hardy type) or the increasing potential (Hénon type) in the nonlinear term. To deal with the Hardy–Hénon type nonlinearities, we employ weighted Lorentz spaces as solution spaces. We prove unconditional uniqueness and non-uniqueness, and we establish uniqueness criterion for Hardy–Hénon parabolic equations in the weighted Lorentz spaces. The results extend the previous works on the Fujita equation and Hardy equations in Lebesgue spaces.

Diffusive stability and self-similar decay for the harmonic map heat flow

In this paper we study the harmonic map heat flow on the euclidean space Rd and we show an unconditional uniqueness result for maps with small initial data in the homogeneous Besov space B˙p,∞dp(Rd) where d<p<∞. As a consequence we obtain decay rates for solutions of the harmonic map flow of the form ‖∇u(t)‖L∞(Rd)≤Ct−12.Additionally, under the assumption of a stronger spatial localization of the initial conditions, we show that the temporal decay happens in a self-similar way. We also explain that similar results hold for the biharmonic map heat flow and the semilinear heat equation with a power-type nonlinearity.

Exact convergence rate of solutions for a semilinear heat equation with a critical and a supercritical exponent revisited

Exact convergence rate of solutions for a semilinear heat equation with a critical and a supercritical exponent revisited

Necessary and sufficient condition for existence for a case of eigenvalues of multiplicity two

We establish necessary and sufficient condition for existence of solutions for a class of semilinear Dirichlet problems with the linear part at resonance at eigenvalues of multiplicity two. The result is applied to give a condition for unboundness of all solutions of the corresponding semilinear heat equation.&#x0D; For more information see https://ejde.math.txstate.edu/Volumes/2024/09/abstr.html

Existence of blowup solutions to the semilinear heat equation with double power nonlinearity

Existence of blowup solutions to the semilinear heat equation with double power nonlinearity

Continuity and pullback attractors for a semilinear heat equation on time-varying domains

We consider dynamics of a semilinear heat equation on time-varying domains with lower regular forcing term. Instead of requiring the forcing term f(cdot ) to satisfy int _{-infty}^{t}e^{lambda s}|f(s)|^{2}_{L^{2}},ds<infty for all tin mathbb{R}, we show that the solutions of a semilinear heat equation on time-varying domains are continuous with respect to initial data in H^{1} topology and the usual (L^{2},L^{2}) pullback mathscr{D}_{lambda}-attractor indeed can attract in the H^{1}-norm, provided that int _{-infty}^{t}e^{lambda s}|f(s)|^{2}_{H^{-1}(mathcal{O}_{s})},ds< infty and fin L^{2}_{mathrm{loc}}(mathbb{R},L^{2}(mathcal{O}_{s})).

Existence and Uniqueness for the Cauchy Problem of Semilinear Heat Equations on Stratified Lie Groups in the Critical Sobolev Space

Existence and Uniqueness for the Cauchy Problem of Semilinear Heat Equations on Stratified Lie Groups in the Critical Sobolev Space

Breakdown of $ C^{\infty} $-smoothing effects of solutions to the semilinear heat equation in the whole space

Breakdown of $ C^{\infty} $-smoothing effects of solutions to the semilinear heat equation in the whole space

Gradient blowup profile for the semilinear heat equation

Gradient blowup profile for the semilinear heat equation