Higher-order Semilinear Parabolic Equation Research Articles

Large time behavior of ODE type solutions to higher-order semilinear parabolic equations with small initial data

Let u be a solution to the Cauchy problem for semilinear parabolic equations(Pm){∂tu+(−Δ)mu=|u|αinRN×(0,∞),u(x,0)=λ+φ(x)>0inRN, where m=1,2,⋯,0<α<1,N≥1,λ>0,φ∈BC(RN)∩Lr(RN) with 1≤r<∞. In the case of m=1, by the comparison principle, one can easily show that the solution u to problem (Pm) behaves like a positive solution to ODE ζ′=ζα in (0,∞) as t→∞. In the case of m≥2, because of the lack of the comparison principle, it is not obvious that the solution u to problem (Pm) behaves like a solution to the ODE as t→∞. In this paper, by imposing a smallness condition‖φ‖L∞(RN)<c with a certain constant c dependent on m,α and λ, we prove that the solution u behaves like a solution to the ODE as t→∞. Furthermore, we obtain the precise description of the large time behavior of the solution u and reveal the relationship between the diffusion effect (Pm) has.

Existence of solutions to nonlinear parabolic equations via majorant integral kernel

We establish the existence of solutions to the Cauchy problem for a large class of nonlinear parabolic equations including fractional semilinear parabolic equations, higher-order semilinear parabolic equations, and viscous Hamilton–Jacobi equations by using the majorant kernel introduced in Ishige et al. (2020).

Blow-up rates for a higher-order semilinear parabolic equation with nonlinear memory term

ABSTRACT In this paper, we establish blow-up rates for a higher-order semilinear parabolic equation with nonlocal in time nonlinearity with no positive assumption on the solution. We also give Liouville-type theorem for higher-order semilinear parabolic equation with infinite memory nonlinear term which plays the main tools to prove our blow-up rate result. Finally, we study the well-posedness of mild solutions.

Existence of solutions for a higher-order semilinear parabolic equation with singular initial data

We establish the existence of solutions of the Cauchy problem for a higher-order semilinear parabolic equation by introducing a new majorizing kernel. We also study necessary conditions on the initial data for the existence of local-in-time solutions and identify the strongest singularity of the initial data for the solvability of the Cauchy problem.

Construction of type I blowup solutions for a higher order semilinear parabolic equation

Abstract We consider the higher-order semilinear parabolic equation $$\begin{array}{} \displaystyle \partial_t u = -(-{\it\Delta})^{m} u + u|u|^{p-1}, \end{array}$$ in the whole space ℝN, where p &gt; 1 and m ≥ 1 is an odd integer. We exhibit type I non self-similar blowup solutions for this equation and obtain a sharp description of its asymptotic behavior. The method of construction relies on the spectral analysis of a non self-adjoint linearized operator in an appropriate scaled variables setting. In view of known spectral and sectorial properties of the linearized operator obtained by Galaktionov [15], we revisit the technique developed by Merle-Zaag [23] for the classical case m = 1, which consists in two steps: the reduction of the problem to a finite dimensional one, then solving the finite dimensional problem by a classical topological argument based on the index theory. Our analysis provides a rigorous justification of a formal result in [15].

Fujita phenomenon in higher-order parabolic equation with nonlocal term

This paper considers the Cauchy problem for a higher-order semilinear parabolic equation with nonlocal term in , where , , , and . We obtain the critical Fujita exponent to the problem, in the sense that every nontrivial solution blows up in finite time whenever , and there are both global and non-global solutions if .

Space-Time Estimates of Mild Solutions of a Class of Higher-Order Semilinear Parabolic Equations in Lp

AbstractWe establish the well-posedness of boundary value problems for a family of nonlinear higherorder parabolic equations which comprises some models of epitaxial growth and thin film theory. In order to achieve this result, we provide a unified framework for constructing local mild solutions in C

Asymptotically self-similar global solutions for a higher-order semilinear parabolic equation

In this paper, we study the higher-order semilinear parabolic equation {u t +(-Δ) m u=a|u| p-1 u, (t,x) ∈ℝ 1 + × ℝ N , u(0,x)=ϕ(x), x∈ℝ N , where m, p>1 and a ∈ ℝ. For p>1 +2m/N, we prove that the global existence of mild solutions for small initial data with respect to some norm. Some of those solutions are proved to be asymptotic self-similar.

Blow-up rates for higher-order semilinear parabolic equations and systems and some Fujita-type theorems

In this paper, we derive blow-up rates for higher-order semilinear parabolic equations and systems. Our proof is by contradiction and uses a scaling argument. This procedure reduces the problems of blow-up rate to Fujita-type theorems. In addition, we also give some new Fujita-type theorems for higher-order semilinear parabolic equations and systems with the time variable on R . These results are not restricted to positive solutions.

On Interfaces and Oscillatory Solutions of Higher-Order Semilinear Parabolic Equations with Non-Lipschitz Nonlinearities

We study local properties of solutions and their asymptotic extinction behavior for the fourth-order semilinear parabolic equation of diffusion-absorption type where p < 1, so that the absorption term is not Lipschitz continuous at u = 0. The Cauchy problem with bounded compactly supported initial data possesses solutions with finite interfaces, and we describe their oscillatory, sign changing properties for p ∈ (-1 3 1). For p ∈ (0, 1), we also study positive solutions of the free-boundary problem with zero contact angle and zero-flux conditions. Finally, we describe families {f k } of similarity extinction patterns u s (x, t) = (T - t) 1/(1-p) f(y), where y = x/(T- t) 1/4 , that vanish in finite time, as t → T - ∈ (0, ∞). Similar local and asymptotic properties are indicated for the sixth-order equation with source u t = u xxxxxx ± |u| p-1 u , where p ∈ (-1 5, 1).

Continuation and asymptotics of solutions to semilinear parabolic equations with critical nonlinearities

In this paper we discuss continuation properties and asymptotic behavior of ɛ-regular solutions to abstract semilinear parabolic problems in case when the nonlinear term satisfies critical growth conditions. A necessary and sufficient condition for global in time existence of ɛ-regular solutions is given. We also formulate sufficient conditions to construct a piecewise ɛ-regular solutions (continuation beyond maximal time of existence for ɛ-regular solutions). Applications to strongly damped wave equations and to higher order semilinear parabolic equations are finally discussed. In particular global solvability and the existence of a global attractor for u t t + η ( − Δ D ) 1 2 u t + ( − Δ D ) u = f ( u ) in H 0 1 ( Ω ) × L 2 ( Ω ) is achieved in case when a nonlinear term f satisfies a critical growth condition and a dissipativeness condition. Similar result is obtained for a 2 mth order semilinear parabolic initial boundary value problem in a Hilbert space H 2 , { B j } m ( Ω ) .

Non-uniqueness and global similarity solutions for a higher-order semilinear parabolic equation

We consider a 2mth-order (m ≥ 2) semilinear parabolic equation of the reaction–diffusion type, and initial data , q ≥ 1. This is a higher-order extension of the classical semilinear heat equation for m = 1 from combustion theory. It is well known from Weissler's results that, for p < p0 = 1 + 2mq/N, there exists a unique local in time solution as a continuous curve for sufficiently small T > 0. For m = 1, it was proved that local nonexistence can happen for p > p0. In the case m = 1, which was studied in greater detail in the 1980s, the precise range p ≤ 1 + 2q/N for uniqueness of such solutions has been established. For p > 1 + 2q/N, non-uniqueness was proved by Haraux and Weissler by constructing special similarity solutions.Our goal is to show that non-uniqueness takes place for the higher-order parabolic equations if p > p0. To this end, we describe a discrete subset of similarity solutions where each V is a radial, exponentially decaying solution of the elliptic equation By perturbation techniques, we establish the existence of radially symmetric similarity profiles Vl for p close to critical bifurcation exponents pl = 1 + 2m/(N + l), l = 0, 2, …, and prove that all the p-bifurcation branches remain in the subcritical Sobolev range p < pS = (N + 2m)/(N − 2m)+. By using analytic, asymptotic and numerical methods we justify some global properties of the bifurcation diagram. We also demonstrate that the similarity profiles satisfy Sturm's zero property in a certain ‘approximate’ sense.

On higher-order semilinear parabolic equations with measures as initial data

We consider 2mth-order (m 2) semilinear parabolic equations ut = ( 1) m u ± |u| p 1 u in R N ◊ R+ (p > 1), with Dirac's mass (x) as the initial function. We show that for p 0, while for p p0 such a local in time solution does not exist. This leads to a boundary layer phenomenon in constructing a proper solution via regular approximations.

On very singular similarity solutions of a higher-order semilinear parabolic equation**Research supported by TMR networks ERB FMRX CT98-0201 and RTN network HPRN-CT-2002-00274.

We study the large-time behaviour of solutions of a semilinear 2mth-order parabolic equation with bounded integrable initial data u0 decaying exponentially at infinity. For the semilinear heat equation (m = 1), the asymptotic behaviour was established in detail in the 1980s. Our main goal is to justify that, for any m ⩾ 1 in the subcritical range 1 < p < p0 = 1 + (2m/N), there exists a finite number, M ∼ N(p0 − p)/2(p − 1) → ∞ as p → 1+, of different very singular self-similar solutions of the form where each V is a radial, exponentially decaying solution of the elliptic equation By a perturbation technique, we establish the existence of radially symmetric very singular solution profiles Vl for p close to critical bifurcation exponents pl = 1 + (2m/(l + N)), l = 0, 2, …, where the first one, V0, is shown to be stable. Discrete and countable subsets of other self-similar and approximately self-similar patterns are introduced.

Self-Similar Blow-Up in Higher-Order Semilinear Parabolic Equations

We study the Cauchy problem in $\re \times \re_+$ for one-dimensional 2mth-order, m>1, semilinear parabolic PDEs of the form ($D_x=\partial/\partial x$) \[ u_t = \Dx u + |u|^{p-1}u, \,\,\,\mbox{where} \,\, \,\,\,p > 1, \quad \mbox{ and } \quad u_t = \Dx u + e^u \] with bounded initial data u0 (x). Specifically, we are interested in those solutions that blow up at the origin in a finite time T. We show that, in contrast to the solutions of the classical second-order parabolic equations ut = uxx + up and ut = uxx + eu from combustion theory, the blow-up in their higher-order counterparts is asymptotically self-similar. In particular, there exist exact nontrivial self-similar blow-up solutions, u* (x,t) = (T-t)-1/(p-1)f (y) in the case of the polynomial nonlinearity and u(x,t) = -ln(T-t) + f(y) for the exponential nonlinearity, where y= x/(T-t)1/2m is the backward higher-order heat kernel variable. The profiles f(y) satisfy related semilinear ODEs that share the same non--self-adjoint higher-order linear differential operators. We show that there are at least $2 \lfloor \frac m2 \rfloor$ nontrivial self-similar solutions to the full PDEs. Numerical solution of the ODEs for m=2 and 3 supports this, and the time dependent solutions of the PDEs for m=2 are then studied by using a scale invariant adaptive numerical method. It is shown that those functions f(y), which have the simplest spatial shape (e.g., a single maximum), correspond to stable self-similar solutions. A further countable subset of nonsimilarity blow-up patterns can be constructed by linearization and matching with similarity solutions of a first-order Hamilton--Jacobi equation.

Global solutions of higher-order semilinear parabolic equations in the supercritical range

We study the asymptotic behaviour of global bounded solutions of the Cauchy problem for the semilinear $2m$-th order parabolic equation $$ u_t = -(-\Delta)^m u + |u|^p \quad {\rm in} \,\,\, {{\bf R}^N} \times \mathbb R_+, $$ where $m>1$, $p>1$, with bounded integrable initial data $u_0$. We prove that, in the supercritical Fujita range $p > p_F = 1+2m/N$, any small global solution with nonnegative initial mass, $\int u_0 dx \ge 0$, exhibits as $t \to \infty$ the asymptotic behaviour given by the fundamental solution of the linear parabolic operator (unlike the case $p \in (1,p_F]$ where solutions blow-up for any arbitrarily small nonnegative nontrivial initial data). A discrete spectrum of other possible asymptotic patterns and the corresponding monotone sequence of critical exponents $\{p_l= 1+2m/(l+N), \, l=0,1,2,...\}$, where $p_0=p_F$, are discussed.

On the asymptotics of global solutions of higher-order semilinear parabolic equations in the supercritical range

We study the asymptotic behaviour of global bounded solutions of the Cauchy problem for the semilinear 2 mth order parabolic equation u t =−(− Δ) m u+| u| p in R N × R +, where m>1, p>1, with bounded integrable initial data u 0. We prove that in the supercritical Fujita range p> p F =1+2 m/ N any small global solution with nonnegative initial mass, ∫u 0 dx⩾0 , exhibits as t→∞ the asymptotic behaviour given by the fundamental solution of the linear parabolic operator (unlike the case p∈ ]1,p F] where solutions can blow-up for any arbitrarily small initial data). A discrete spectrum of other possible asymptotic patterns and the corresponding monotone sequence of critical exponents {p l=1+2m/(l+N), l=0,1,2,…} , where p 0= p F , are discussed. To cite this article: Yu.V. Egorov et al., C. R. Acad. Sci. Paris, Ser. I 335 (2002) 805–810.

Existence and blow-up for higher-order semilinear parabolic equations: Majorizing order-preserving operators

As a basic example, we establish that in the Cauchy problem for the 2m-th order semilinear parabolic equation u t = -(-Δ) m u + |u| P , x ∈ R n , t > 0; u(x, 0) = u 0 (x), x ∈ R n , where m > 1, p > 1, with bounded integrable initial data u 0 , the critical Fujita exponent is p F = 1 +2m/N, so that for p > p F there exists a class of small global solutions, and for p ∈ (1, p F ] blow-up can occur for arbitrarily small initial data. The analysis of the asymptotics of both classes of global and blow-up solutions is based on comparison with similarity solutions of the majorizing order-preserving integral equation, which is shown to exist for any m > 1. Generalizations to differential and integral evolution equations and relations to positivity sets for higher-order equations are discussed.

Regional blow-up for a higher-order semilinear parabolic equation

We study the blow-up behaviour of solutions of a 2mth order semilinear parabolic equation[formula here]with a superlinear function q(u) for |u| Gt; 1. We prove some estimates on the asymptotic blow-up behaviour. Such estimates apply to general integral evolution equations. We answer the following question: find a continuous function q(u) with a superlinear growth as u → ∞ such that the parabolic equation exhibits regional blow-up in a domain of finite non-zero measure. We show that such a regional blow-up can occur for q(u) = u|ln|u‖2m. We present a formal asymptotic theory explaining that the stable (generic) blow-up behaviour as t → T− is described by the self-similar solution[formula here]of the complex Hamilton–Jacobi equation[formula here].

Local solvability of higher order semilinear parabolic equations

Local solvability of higher order semilinear parabolic equations