Large time behavior for a viscous Hamilton-Jacobi equation with Neumann boundary condition

We prove the existence and the uniqueness of strong solutions for the viscous Hamilton-Jacobi equation: with Neumann boundary condition, and initial data μ₀, a continuous function. The domain Ω is a bounded and convex open set with smooth boundary, a∈R,a≠0 and p>0. Then, we study the large time behavior of the solution and we show that for p∈(0,1), the extinction in finite time of the gradient of the solution occurs, while for p?1 the solution converges uniformly to a constant, as t→∞.     

Global solutions and self-similar solutions of semilinear parabolic equations with nonlinear gradient terms

In this paper we consider the Cauchy problem of semilinear parabolic equations with nonlinear gradient terms a(x)|u|^q−1u|∇u|^p. We prove the existence of global solutions and self-similar solutions for small initial data. Moreover, for a class of initial data we show that the global solutions behave asymptotically like self-similar solutions as t→∞.     

An elliptic system with bifurcation parameters on the boundary conditions

In this paper we consider the elliptic system Δu=a(x)upvq, Δv=b(x)urvs in Ω, a smooth bounded domain, with boundary conditions , on ∂Ω. Here λ and μ are regarded as parameters and p,s>1, q,r>0 verify (p−1)(s−1)>qr. We consider the case where a(x)?0 in Ω and a(x) is allowed to vanish in an interior subdomain Ω₀, while b(x)>0 in . Our main results include existence of nonnegative nontrivial solutions in the range 0<λ<λ₁?∞, μ>0, where λ₁ is characterized by means of an eigenvalue problem, and the uniqueness of such solutions. We also study their asymptotic behavior in all possible cases: as both λ,μ→0, as λ→λ₁<∞ for fixed μ (respectively μ→∞ for fixed λ) and when both λ,μ→∞ in case λ₁=∞.     

Carleson measure problems for parabolic Bergman spaces and homogeneous Sobolev spaces

Let be the space of solutions to the parabolic equation having finite norm. We characterize nonnegative Radon measures μ on having the property , 1≤p≤q<∞, whenever . Meanwhile, denoting by v(t,x) the solution of the above equation with Cauchy data v₀(x), we characterize nonnegative Radon measures μ on satisfying , β∈(0,n), p∈[1,n/β], q∈(0,∞). Moreover, we obtain the decay of v(t,x), an isocapacitary inequality and a trace inequality.     

Universal bounds for global solutions of a diffusion equation with a nonlocal reaction term

We consider the nonlinear heat equation with nonlocal reaction term in space , in smoothly bounded domains. We prove the existence of a universal bound for all nonnegative global solutions of this equation. Moreover, in contrast with similar recent results for equations with local reaction terms, this is shown to hold for all p>1. As an interesting by-product of our proof, we derive for this equation a smoothing effect under weaker assumptions than for corresponding problem with local reaction.     

Positive solutions of the p-Laplacian involving a superlinear nonlinearity with zeros

Using a combination of several methods, such as variational methods, the sub and supersolutions method, comparison principles and a priori estimates, we study existence, multiplicity, and the behavior with respect to λ of positive solutions of p-Laplace equations of the form −Δpu=λh(x,u), where the nonlinear term has p-superlinear growth at infinity, is nonnegative, and satisfies h(x,a(x))=0 for a suitable positive function a. In order to manage the asymptotic behavior of the solutions we extend a result due to Redheffer and we establish a new Liouville-type theorem for the p-Laplacian operator, where the nonlinearity involved is superlinear, nonnegative, and has positive zeros.     

Asymptotic analysis of the p-Laplacian flow in an exterior domain

We consider the Dirichlet problem for the p  -Laplacian evolution equation, ut=Δpu$u_t={\mathrm{\Delta }}_{p}u$, where p>2$p>2$, posed in an exterior domain in RN${\mathbf{R}}^N$, with zero Dirichlet boundary condition and with integrable and nonnegative initial data. We are interested in describing the influence of the holes of the domain on the large time behaviour of the solutions. Such behaviour varies depending on the relative values of N and p  . We must distinguish between the behaviour near infinity of space (outer analysis), and near the holes (inner analysis). We obtain that the outer analysis is given in all cases by certain self-similar solutions and the inner analysis is given by quasi-stationary states. Logarithmic corrections to exact self-similarity appear in the critical case N=p$N=p$, which is mathematically more interesting. In this first paper we treat only the cases N>p$N>p$ and N=p$N=p$, the case N<p$N<p$ will be considered in a companion work.     

Blowup stability of solutions of the nonlinear heat equation with a large life span

We study the Cauchy problem for the nonlinear heat equation u_t-?u=|u|^p-1u in R^N. The initial data is of the form u₀=λ?, where ?∈C₀(R^N) is fixed and λ>0. We first take 1<p<p_f, where p_f is the Fujita critical exponent, and ?∈C₀(R^N)∩L¹(R^N) with nonzero mean. We show that u(t) blows up for λ small, extending the H. Fujita blowup result for sign-changing solutions. Next, we consider 1<p<p_s, where p_s is the Sobolev critical exponent, and ?(x) decaying as |x|^-σ at infinity, where p<1+2/σ. We also prove that u(t) blows up when λ is small, extending a result of T. Lee and W. Ni. For both cases, the solution enjoys some stable blowup properties. For example, there is single point blowup even if ? is not radial.     

Approximate controllability for nonlinear degenerate parabolic problems with bilinear control

In this paper, we study the global approximate multiplicative controllability for nonlinear degenerate parabolic Cauchy–Neumann problems. First, we obtain embedding results for weighted Sobolev spaces, that have proved decisive in reaching well-posedness for nonlinear degenerate problems. Then, we show that the above systems can be steered in L²$L^2$ from any nonzero, nonnegative initial state into any neighborhood of any desirable nonnegative target-state by bilinear piecewise static controls. Moreover, we extend the above result relaxing the sign constraint on the initial data.     

Local smoothing effects, positivity, and Harnack inequalities for the fast p-Laplacian equation

We study qualitative and quantitative properties of local weak solutions of the fast p-Laplacian equation, t_∂u=Δpu, with 1<p<2. Our main results are quantitative positivity and boundedness estimates for locally defined solutions in domains of Rn×[0,T]. We combine these lower and upper bounds in different forms of intrinsic Harnack inequalities, which are new in the very fast diffusion range, that is when 1<p?2n/(n+1). The boundedness results may be also extended to the limit case p=1, while the positivity estimates cannot.We prove the existence as well as sharp asymptotic estimates for the so-called large solutions for any 1<p<2, and point out their main properties.We also prove a new local energy inequality for suitable norms of the gradients of the solutions. As a consequence, we prove that bounded local weak solutions are indeed local strong solutions, more precisely .     

Positivity, local smoothing, and Harnack inequalities for very fast diffusion equations

We investigate qualitative properties of local solutions u(t,x)?0 to the fast diffusion equation, t_∂u=Δ(um)/m with m<1, corresponding to general nonnegative initial data. Our main results are quantitative positivity and boundedness estimates for locally defined solutions in domains of the form [0,T]×Ω, with Ω⊆Rd. They combine into forms of new Harnack inequalities that are typical of fast diffusion equations. Such results are new for low m in the so-called very fast diffusion range, precisely for all m?mc=(d−2)/d. The boundedness statements are true even for m?0, while the positivity ones cannot be true in that range.     

Self-similar singular solutions of a p-Laplacian evolution equation with absorption

We consider, for p∈(1,2) and q>1, self-similar singular solutions of the equation v_t=div(|∇v|^p−2∇v)−v^q in Rn×(0,∞); here by self-similar we mean that v takes the form v(x,t)=t^−αw(|x|t^−αβ) for α=1/(q−1) and β=(q+1−p)/p, whereas singular means that v is non-negative, non-trivial, and for all x≠0. That is, we consider the ODE problem

(0.1)     

Blow-up of solutions to a degenerate parabolic equation not in divergence form

We study nonglobal positive solutions to the Dirichlet problem for u_t=u^p(Δu+u) in bounded domains, where 0<p<2. It is proved that the set of points at which u blows up has positive measure and the blow-up rate is exactly . If either the space dimension is one or p<1, the ω-limit set of consists of continuous functions solving . In one space dimension it is shown that actually as t→T, where w coincides with an element of a one-parameter family of functions inside each component of its positivity set; furthermore, we study the size of the components of {w>0} with the result that this size is uniquely determined by Ω in the case p<1, while for p>1, the positivity set can have the maximum possible size for certain initial data, but it may also be arbitrarily close to the minimal length π.     

Asymptotic behavior for nonlocal dispersal equations

This paper is concerned with the existence and asymptotic behavior of solutions of a nonlocal dispersal equation. By means of super-subsolution method and monotone iteration, we first study the existence and asymptotic behavior of solutions for a general nonlocal dispersal equation. Then, we apply these results to our equation and show that the nonnegative solution is unique, and the behavior of this solution depends on parameter λ in equation. For λ≤λ₁(Ω), the solution decays to zero as t→∞; while for λ>λ₁(Ω), the solution converges to the unique positive stationary solution as t→∞. In addition, we show that the solution blows up under some conditions.     

On the asymptotic behavior of unbounded radial solutions for semilinear parabolic problems involving critical Sobolev exponent

This paper is concerned with a semilinear parabolic equation involving critical Sobolev exponent in a ball or in RN. The asymptotic behavior of unbounded, radially symmetric, nonnegative global solutions which do not decay to zero is given. The structure of the space of initial data is also discussed.     

Asymptotic stability for a free boundary problem arising in a tumor model

We consider a tumor model in which all cells are proliferating at a rate μ and their density is proportional to the nutrient concentration. The model consists of a coupled system of an elliptic equation and a parabolic equation, with the tumor boundary as a free boundary. It is known that for an appropriate choice of parameters, there exists a unique spherically symmetric stationary solution with radius RS which is independent of μ. It was recently proved that there is a function μ_∗(RS) such that the spherical stationary solution is linearly stable if μ<μ_∗(RS) and linearly unstable if μ>μ_∗(RS). In this paper we prove that the spherical stationary solution is nonlinearly stable (or, asymptotically stable) if μ<μ_∗(RS).     

Global strong solution to the semi-linear Keller-Segel system of parabolic-parabolic type with small data in scale invariant spaces

We shall show existence of global strong solution to the semi-linear Keller-Segel system in Rn, n?3, of parabolic-parabolic type with small initial data and for max{1,n/4}<r<n/2. Our method is based on the perturbation of linealization together with the Lp-Lq estimates of the heat semigroup and the fractional powers of the Laplace operator. As a by-product of our method, we shall prove the decay property of solutions as the time goes to infinity.     

Second critical exponent and life span for pseudo-parabolic equation

This paper studies the second critical exponent and life span of solutions for the pseudo-parabolic equation u_t−kΔu_t=Δu+u^p in Rⁿ×(0,T), with p>1, k>0. It is proved that the second critical exponent, i.e., the decay order of the initial data required by global solutions in the coexistence region of global and non-global solutions, is independent of the pseudo-parabolic parameter k. Nevertheless, it is revealed that the viscous term kΔu_t relaxes restrictions on the amplitude of the initial data required by the global solutions. Moreover, it is observed that the life span of the non-global solutions will be delayed by the third order viscous term. Finally, some numerical examples are given to illustrate all these results.     

Universal self-similarity of porous media equation with absorption: the critical exponent case

In this paper we study the large time behavior of non-negative solutions to the Cauchy problem of u_t=Δu^m−u^q in R^N×(0,∞), where m>1 and q=q_c≡m+2/N is a critical exponent. For non-negative initial value u(x,0)=u₀(x)∈L¹(R^N), we show that the solution converges, if u₀(x)(1+|x|)^k is bounded for some k>N, to a unique fundamental solution of u_t=Δu^m, independent of the initial value, with additional logarithmic anomalous decay exponent in time as t→∞.