共查询到20条相似文献,搜索用时 62 毫秒
1.
We consider a class of ultraparabolic differential equations that satisfy the Hörmander’s hypoellipticity condition and we prove that the weak solutions to the equation with measurable coefficients are locally bounded functions. The method extends the Moser’s iteration procedure and has previously been employed in the case of operators verifying a further homogeneity assumption. Here we remove that assumption by proving some potential estimates and some ad hoc Sobolev type inequalities for solutions. 相似文献
2.
Qiuyi Dai Yonggeng Gu Jiuyi Zhu 《Nonlinear Analysis: Theory, Methods & Applications》2011,74(18):7126-7136
This paper concerns a priori estimates and existence of solutions of generalized mean curvature equations with Dirichlet boundary value conditions in smooth domains. Using the blow-up method with the Liouville-type theorem of the p laplacian equation, we obtain a priori bounds and the estimates of interior gradient for all solutions. The existence of positive solutions is derived by the topological method. We also consider the non-existence of solutions by Pohozaev identities. 相似文献
3.
Guowei LIU 《数学物理学报(B辑英文版)》2017,37(1):79-96
This paper studies the time asymptotic behavior of solutions for a nonlinear convection diffusion reaction equation in one dimension.First,the pointwise estimates of solutions are obtained,furthermore,we obtain the optimal L~p,1≤ p ≤ +∞,convergence rate of solutions for small initial data.Then we establish the local existence of solutions,the blow up criterion and the sufficient condition to ensure the nonnegativity of solutions for large initial data.Our approach is based on the detailed analysis of the Green function of the linearized equation and some energy estimates. 相似文献
4.
Weike Wang 《Journal of Differential Equations》2003,187(2):310-336
By introducing a new approximate Green function, we obtain the pointwise estimates on the solutions of Euler equations with linear frictional damping, from which we can deduce the optimal convergence rates to the nonlinear diffusion waves. The pointwise estimates and convergence rates given in this paper are new. 相似文献
5.
Yongzhong Wang Pengcheng NiuXuewei Cui 《Nonlinear Analysis: Theory, Methods & Applications》2011,74(17):6265-6286
In this paper, based on measure theoretical arguments, we establish Harnack estimates and Hölder continuity of nonnegative weak solutions for a degenerate parabolic equation with a singular weight. We transform the equation by performing the change of function. The energy estimates, the upper boundedness, the lower boundedness and the expansion of positivity for the solutions to the transformed equation are obtained. Then our aim is reached. 相似文献
6.
This paper deals with parabolic equation ut=Δu+r|∇u|−aepu subject to nonlinear boundary flux ∂u/∂η=equ, where r>1, p,q,a>0. There are two positive sources (the gradient reaction and the boundary flux) and a negative one (the absorption) in the model. It is well known that blow-up or not of solutions depends on which one dominating the model, the positive or negative sources, and furthermore on the absorption coefficient for the balance case of them. The aim of the paper is to study the influence of the reactive gradient term on the asymptotic behavior of solutions. We at first determine the critical blow-up exponent, and then obtain the blow-up rate, the blow-up set as well as the spatial blow-up profile for blow-up solutions in the one-dimensional case. It turns out that the gradient term makes a substantial contribution to the formation of blow-up if and only if r?2, where the critical r=2 is such a balance situation of the two positive sources for which the effects of the gradient reaction and the boundary source are at the same level. In addition, it is observed that the gradient term with r>2 significantly affects the blow-up rate also. In fact, the gained blow-up rates themselves contain the exponent r of the gradient term. Moreover, the blow-up rate may be discontinuous with respect to parameters included in the problem due to convection. As for the influence of gradient perturbations on spatial blow-up profiles, we only need some coefficients related to r for the profile estimates, while the exponent of the profile itself is r-independent. This seems natural for boundary blow-up solutions that the spatial profiles mainly rely on the exponent of the boundary singularity. 相似文献
7.
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 . 相似文献
8.
For the case of multidimensional viscous conservation laws with fourth-order smoothing only, we develop detailed pointwise estimates on the Green's function for the linear fourth-order convection equation that arises upon linearization of the conservation law about a viscous planar wave solution. As in previous analyses in the case of second-order smoothing, our estimates are sufficient to establish that spectral stability implies nonlinear stability, though the full development of this result will be considered in a companion paper. 相似文献
9.
This paper is concerned with distributed and Dirichlet boundary controls of semilinear parabolic equations, in the presence
of pointwise state constraints. The paper is divided into two parts. In the first part we define solutions of the state equation
as the limit of a sequence of solutions for equations with Robin boundary conditions. We establish Taylor expansions for solutions
of the state equation with respect to perturbations of boundary control (Theorem 5.2). For problems with no state constraints,
we prove three decoupled Pontryagin's principles, one for the distributed control, one for the boundary control, and the last
one for the control in the initial condition (Theorem 2.1). Tools and results of Part 1 are used in the second part to derive
Pontryagin's principles for problems with pointwise state constraints.
Accepted 12 July 2001. Online publication 21 December 2001. 相似文献
10.
The Cauchy problem for singularly perturbed parabolic equations is considered, and weighted L2-estimates as well as certain decay properties of bounded classical solutions to it are established. These do not depend on
the value of the small perturbation parameter, and allow to prove global in time existence of strong solutions to certain
boundary-value problems for ultraparabolic equations with unbounded coefficients. Optimal decay estimates are proved for such solutions. All results concerning ultraparabolic equations apply, in particular, to the
Kolmogorov equation for diffusion with inertia, to the (linear) Fokker-Planck equation, to the linearized Boltzmann equation,
and to some nonlinear integro-differential ultraparabolic equations of the Fokker-Planck type, arising from biophysics. Optimal decay estimates
are derived for global in time strong solutions to such equations. 相似文献
11.
Jiebao Sun Jingxue YinYifu Wang 《Nonlinear Analysis: Theory, Methods & Applications》2011,74(6):2415-2424
This paper is concerned with a doubly degenerate parabolic equation with logistic periodic sources. We are interested in the discussion of the asymptotic behavior of solutions of the initial-boundary value problem. In this paper, we first establish the existence of non-trivial nonnegative periodic solutions by a monotonicity method. Then by using the Moser iterative method, we obtain an a priori upper bound of the nonnegative periodic solutions, by means of which we show the existence of the maximum periodic solution and asymptotic bounds of the nonnegative solutions of the initial-boundary value problem. We also prove that the support of the non-trivial nonnegative periodic solution is independent of time. 相似文献
12.
The behavior of solutions to the biharmonic equation is well-understood in smooth domains. In the past two decades substantial
progress has also been made for the polyhedral domains and domains with Lipschitz boundaries. However, very little is known
about higher order elliptic equations in the general setting.
In this paper we introduce new integral identities that allow to investigate the solutions to the biharmonic equation in an
arbitrary domain. We establish:
(1) boundedness of the gradient of a solution in any three-dimensional domain;
(2) pointwise estimates on the derivatives of the biharmonic Green function;
(3) Wiener-type necessary and sufficient conditions for continuity of the gradient of a solution.
Mathematics Subject Classification (2000) 35J40, 35J30, 35B65 相似文献
13.
Lawrence C. Evans Ovidiu Savin 《Calculus of Variations and Partial Differential Equations》2008,32(3):325-347
We propose a new method for showing C
1, α
regularity for solutions of the infinity Laplacian equation and provide full details of the proof in two dimensions. The
proof for dimensions n ≥ 3 depends upon some conjectured local gradient estimates for solutions of certain transformed PDE.
LCE is supported in part by NSF Grant DMS-0500452. OS was supported in part by the Miller Institute for Basic Research in
Science, Berkeley. 相似文献
14.
We establish existence and pointwise estimates of fundamental solutions and Green’s matrices for divergence form, second order strongly elliptic systems in a domain $\Omega \subseteq {\mathbb{R}}^n, n \geq 3We establish existence and pointwise estimates of fundamental solutions and Green’s matrices for divergence form, second order
strongly elliptic systems in a domain , under the assumption that solutions of the system satisfy De Giorgi-Nash type local H?lder continuity estimates. In particular,
our results apply to perturbations of diagonal systems, and thus especially to complex perturbations of a single real equation. 相似文献
15.
We consider the porous medium equation with sign changes. In particular this equation describes the mixing of fresh and salt groundwater due to mechanical dispersion. The unknown function u, which denotes the velocity of the fluids, may take positive as well as negative values. Our main result is the following : under certain monotonicity hypotheses on the initial function, there exists a time T> 0 after which the regions where u < 0 and u > 0 are separated by an interface x = ζ(t) such that ζ is continuously differentiable on [T,∞]. The method of proof is based on a priori estimates for solutions of regularized problems and for their level lines 相似文献
16.
G. Wolansky 《Monatshefte für Mathematik》2001,132(3):255-261
We compare the solution of to the solution of the same equation where f is replaced by a “concentrated” source . As a result we derive some estimates on the solution in spatial norm, locally uniformly in t, with respect to the norm of for any integer . In the case we obtain a critical inequality relating the norm of to an exponential norm of u.
(Received 1 September 2000; in revised form 17 January 2001) 相似文献
17.
Jan Čermák 《Georgian Mathematical Journal》1995,2(1):1-8
The aim of this paper is to find the class of continuous pointwise transformations (as general as possible) in the framework of which Kummer's transformationz(t)=g(t)y(h(t)) represents the most general pointwise transformation converting every linear homogeneous differential equation of thenth order into an equation of the same type. Further, some forms of these equations having certain subspaces of solutions are considered. 相似文献
18.
Jaeun Ku 《BIT Numerical Mathematics》2010,50(3):609-630
Highly localized pointwise error estimates for a stabilized Galerkin method are provided for second-order non-selfadjoint
elliptic partial differential equations. The estimates show a local dependence of the error on the derivative of the solution
u and weak dependence on the global norm. The results in this paper are an extension of the previous pointwise error estimates
for the self-adjoint problems. In order to provide pointwise error estimates in the presence of the first-order term in the
differential equations, we prove that the stabilized Galerkin solution is higher order perturbation to the Ritz projection
of the true solutions. Then, we proceed to obtain pointwise estimates using the so-called discrete Green’s function. Application
to error expansion inequalities and a posteriori error estimators are briefly discussed. 相似文献
19.
We consider a one-dimensional semilinear parabolic equation , for which the spatial derivative of solutions becomes unbounded in finite time while the solutions themselves remain bounded. We establish estimates of blowup rate upper and lower bounds. We prove that in this case the blowup rate does not match the one obtained by the rescaling method. 相似文献
20.
In this paper, we study a fourth order parabolic equation with nonlinear principal part modeling epitaxial thin film growth in two space dimensions. On the basis of the Schauder type estimates and Campanato spaces, we prove the global existence of classical solutions. 相似文献