We are concerned with nonexistence results for a class of quasilinear parabolic differential problems with a potential in $Ω\times(0,+\infty)$, where $Ω$ is a bounded domain. In particular, we investigate how the behavior of the potential near the boundary of the domain and the power nonlinearity affect the nonexistence of solutions. Particular attention is devoted to the special case of the semilinear parabolic problem, for which we show that the critical rate of growth of the potential near the boundary ensuring nonexistence is sharp.
We prove nonexistence of nontrivial, possibly sign changing, stable solutions to a class of quasilinear elliptic equations with a potential on Riemannian manifolds, under suitable weighted volume growth conditions on geodesic balls.
In this article we consider a variational problem related to a quasilinear singular problem and obtain a nonexistence result in a metric measure space with a doubling measure and a Poincaré inequality. Our method is purely variational and to the best of our knowledge, this is the first work concerning singular problems in a general metric setting.
In the present paper we deal with a quasilinear problem involving a singular term and a parametric superlinear perturbation. We are interested in the existence, nonexistence and multiplicity of positive solutions as the parameter $λ>0$ varies. In our first result, the superlinear perturbation has an arbitrary growth and we obtain the existence of a solution for the problem by using the sub-supersolution method. For the second result, the superlinear perturbation has subcritical growth and we employ the Mountain Pass Theorem to show the existence of a second solution.
In this paper, by using the theory of compensated compactness coupled with the kinetic formulation by Lions, Perthame, Souganidis and Tadmor \cite{LPT,LPS}, we prove the existence and nonexistence of global generalized (nonnegative) solutions of the nonlinearly degenerate wave equations $v_{tt} =c (|v|^{s-1} v)_{xx}$ with the nonnegative initial data $v_{0}(x)$ and $ s > 1$. This result is an extension of the results in the second author's paper \cite{Su}, where the existence and the nonexistence of the unique global classical solution were studied with a threshold on $\int_{-\infty}^{\infty} …
We show the existence of nodal solutions to perturbed quasilinear elliptic equations with critical Sobolev exponent on compact Riemannian manifolds. A nonexistence result is also given.
An analogy of nonexistence result by Baras and Goldstein (1984), for the heat equation with inverse singular potential, is proved for 2mth-order linear parabolic equations with Hardy-supercritical singular potentials. Extensions to other linear and nonlinear singular PDEs are discussed.
We are concerned with nonexistence results of nonnegative weak solutions for a class of quasilinear parabolic problems with a potential on complete noncompact Riemannian manifolds. In particular, we highlight the interplay between the geometry of the underlying manifold, the power nonlinearity and the behavior of the potential at infinity.
We provide sufficient and almost optimal conditions for global existence of classical solutions in parabolic Hölder spaces to quasilinear one-dimensional parabolic problems with dynamical boundary conditions.
We introduce a notion of quasilinear parabolic equations over metric measure spaces. Under sharp structural conditions, we prove that local weak solutions are locally bounded and satisfy the parabolic Harnack inequality. Applications include the parabolic maximum principle and pointwise estimates for weak solutions.
Finite time blow-up is shown to occur for solutions to a one-dimensional quasilinear parabolic-parabolic chemotaxis system as soon as the mean value of the initial condition exceeds some threshold value. The proof combines a novel identity of virial type with the boundedness from below of the Liapunov functional associated to the system, the latter being peculiar to the one-dimensional setting.
We establish a spatial gradient maximum principle for classical solutions to the initial and Neumann boundary value problem of some quasilinear parabolic equations on smooth convex domains.
In this paper we prove a nonexistence result for nonlinear parabolic problems with zero lower order term whose model is $$ \begin{cases} u_{t}- Δ_p u+|u|^{q-1}u=λ& \text{in}\ (0,T)\timesΩ u(0,x)=0 & \text{in}\ Ω,\\ u(t,x)=0 & \text{on}\ (0,T)\times\partialΩ, \end{cases} $$ where $Δ_p ={\rm div }(|\nabla u|^{p-2}\nabla u)$ is the usual $p$-laplace operator, $λ$ is measure concentrated on a set of zero parabolic $r$-capacity ($1<p<r$), and $q$ is large enough.
We study linear nonautonomous parabolic systems with dynamic boundary conditions. Next, we apply these results to show a theorem of local existence and uniqueness of a classical solution to a second order quasilinear system with nonlinear dynamic boundary conditions.
Bounded smooth solutions of the Dirichlet and Neumann problems for a wide variety of quasilinear parabolic equations, including graphical anisotropic mean curvature flows, have gradient bounded in terms of oscillation and elapsed time.
Regimes with a singular peaking for a wide class of quasilinear second order parabolic equations are studied. On the basis of energy methods, precise estimates of a final profile of a weak solution in a neighborhood of the peaking time are established depending on the rate of increase of a global energy of this solution. Key words: quasilinear parabolic equations, peaking regimes, energy solutions.
We are concerned with the nonexistence of sign-changing global weak solutions for a class of semilinear parabolic differential inequalities with convection terms in exterior domains. A weight function of the form $t^α|x|^σ$ is considered in front of the power nonlinearity. Two types of non-homogeneous boundary conditions are investigated: Neumann-type and Dirichlet-type boundary conditions. Using a unified approach, for each case, we establish sufficient criteria for the nonexistence of global weak solutions. When $α=0$, the critical exponent in the sense of Fujita is obtained. This exponent i…
We study fractional parabolic equations with indefinite nonlinearities $$ \frac{\partial u} {\partial t}(x,t) +(-Δ)^s u(x,t)= x_1 u^p(x, t),\,\, (x, t) \in \mathbb{R}^n \times \mathbb{R}, $$ where $0<s<1$ and $1<p<\infty$. We first prove that all positive bounded solutions are monotone increasing along the $x_1$ direction. Based on this we derive a contradiction and hence obtain non-existence of solutions. These monotonicity and nonexistence results are crucial tools in a priori estimates and complete blow-up for fractional parabolic equations in bounded domains. To this end, we introduce seve…
Based on a comparison principle, we derive an exponential rate of convergence for solutions to the initial-boundary value problem for a class of quasilinear parabolic equations in one space dimension. We then apply the result to some models in population dynamics and image processing.
In this paper, we provide a direct approach to the existence and uniqueness of strong (in the probabilistic sense) and weak (in the PDE sense) solutions to quasilinear stochastic partial differential equations, which are neither monotone nor locally monotone.
Unboundedness of solutions is shown to occur in a one-dimensional quasilinear parabolicparabolic chemotaxis system for any initial mass. Our result is also independent of the relation between the speeds of the diffusion of cells and chemoattractant. The proof is achieved by contradiction. It uses a virial type inequality together with a boundedness from below of the Liapunov functional associated to the system. Moreover, the same method is applied also in higher dimensions and an unboundedness result is shown in the case of critical quasilinear fully parabolic Keller-Segel system for mass larg…
We study the following quasilinear elliptic equation $$ -Δ_p u + (βΦ(x)-a(x)) u^{p-1} + b(x)g(u)=0 \quad \text{in } \mathbb{R}^N \quad \quad \quad (P_β) $$ where $p>1$, $Φ\in L^\infty_{loc}(\mathbb{R}^N)$, $a,b \in L^\infty(\mathbb{R}^N)$, $β,b,g \geq 0$. We provide a sharp criterion in terms of generalized principal eigenvalue for existence/nonexistence of positive solutions of ($P_β$) in suitable classes of functions. We derive the uniqueness result of ($P_β$) in those classes. Under additional conditions on $Φ$, we further show that : i) either for every $β\geq 0$ nonexistence phenomenon oc…
Asymptotic solutions of a quasilinear parabolic equation with a small parameter at the higher derivative are constructed near large-gradient and Lagrange singularities of A-type, which represent interest for studying processes of shock waves formation in physical media with a small nonzero viscosity.
We establish global regularity for weak solutions to quasilinear divergence form elliptic and parabolic equations over Lipschitz domains with controlled growth conditions on low order terms. The leading coefficients belong to the class of BMO functions with small mean oscillations with respect to $x$.
Motivated by the study of conditional stability of traveling waves, we give an elementary $H^2$ center stable manifold construction for quasilinear parabolic PDE, sidestepping apparently delicate regularity issues by the combination of a carefully chosen implicit fixed-point scheme and "damping-type" $H^s$ energy estimates of a type familiar from the study of hyperbolic--parabolic and relaxation systems. An important feature of these methods is that they generalize to situations such as the hyperbolic--parabolic or relaxation case for which parabolic-type smoothing estimates are unavailable. A…
We present in this note a local in time well-posedness result for the singular $2$-dimensional quasilinear generalized parabolic Anderson model equation $$ \partial_t u - a(u)Δu = g(u)ξ$$ The key idea of our approach is a simple transformation of the equation which allows to treat the problem as a semilinear problem. The analysis is done within the elementary setting of paracontrolled calculus.
We bound the modulus of continuity of solutions to quasilinear parabolic equations in one space variable in terms of the initial modulus of continuity and elapsed time. In particular we characterize those equations for which the Lipschitz constants of solutions can be bounded in terms of their initial oscillation and elapsed time.
We consider a class of abstract quasilinear parabolic problems with lower--order terms exhibiting a prescribed singular structure. We prove well--posedness and Lipschitz continuity of associated semiflows. Moreover, we investigate global existence of solutions and we extend the generalized principle of linearized stability to settings with initial values in critical spaces. These general results are applied to the surface diffusion flow in various settings.
This paper is concerned with a quaslinear parabolic equation including a nonlinear nonlocal initial condition. The problem arises as equilibrium equation in population dynamics with nonlinear diffusion. We make use of global bifurcation theory to prove existence of an unbounded continuum of positive solutions.
In this paper we develop a geometric theory for quasilinear parabolic problems in weighted $L_p$-spaces. We prove existence and uniqueness of solutions as well as the continuous dependence on the initial data. Moreover, we make use of a regularization effect for quasilinear parabolic equations to study the $\om$-limit sets and the long-time behaviour of the solutions. These techniques are applied to a free boundary value problem. The results in this paper are mainly based on maximal regularity tools in (weighted) $L_p$-spaces.
We discuss the existence and uniqueness of the weak solution of the following quasilinear parabolic equation $u_t-Δ_{p(x)}u = f(x,u)$ in $ (0,T)\timesΩ$; $u = 0$ on $(0,T)\times\partialΩ$; $u(0,x)=u_0(x)$ in $Ω$; involving the $p(x)$-Laplacian operator. Next, we discuss the global behaviour of solutions and in particular some stabilization properties.
In this paper we prove finite-time blowup of radially symmetric solutions to the quasilinear parabolic-parabolic two-dimensional Keller-Segel system for any positive mass. This is done in case of nonlinear diffusion and also in the case of nonlinear cross-diffusion provided the nonlinear chemosensitivity term is assumed not to decay. Moreover, it is shown that the above-mentioned lack of non-decay assumption is essential with respect to keeping the dichotomy finite-time blowup against boundedness of solutions. Namely, we prove that without the non-decay assumption possible asymptotic behaviour…
In this paper, we prove global weighted Lorentz and Lorentz-Morrey estimates for gradients of solutions to the quasilinear parabolic equations: $$u_t-\operatorname{div}(A(x,t,\nabla u))=\operatorname{div}(F),$$ in a bounded domain $Ω\times (0,T)\subset\mathbb{R}^{N+1}$, under minimal regularity assumptions on the boundary of domain and on nonlinearity $A$. Then results yields existence of a solution to the Riccati type parabolic equations: $$u_t-\operatorname{div}(A(x,t,\nabla u))=|\nabla u|^q+\operatorname{div}(F)+μ,$$ where $q>1$ and $μ$ is a bounded Radon measure.
In this paper we study a rather wide class of quasilinear parabolic problems with nonlinear boundary condition and nonstandard growth terms. It includes the important case of equations with a $p(t,x)$-Laplacian. By means of the localization method and De Giorgi's iteration technique we derive global a priori bounds for weak solutions of such problems. Our results seem to be new even in the constant exponent case.
In this paper, we are interested in obtaining a unified approach for $C^{1,α}$ estimates for weak solutions of quasilinear parabolic equations, the prototype example being \[ u_t - \text{div} (|\nabla u|^{p-2} \nabla u) = 0. \] without having to consider the singular and degenerate cases separately. This is achieved via a new scaling and a delicate adaptation of the covering argument developed by E.~DiBenedetto and A.~Friedman.
We study unbounded (viscosity) supersolutions of the Evolutionary p-Laplace Equation in the slow diffusion case. The supersolutions fall into two widely different classes, depending on whether they are locally summable to the power p-2 or not. Also the Porous Medium Equation is studied.
This paper studies the existence, nonexistence and uniqueness of positive solutions for a class of quasilinear equations. We also analyze the behavior of this solutions with respect to two parameters $κ$ and $λ$ that appears in the equation. The proof of our main results relies on bifurcation techniques, the sub and supersolution method and a construction of an appropriate large solutions.
This work deals with existence of solutions for the class of quasilinear elliptic problems with cylindrical singularities and multiple critical nonlinearities that can be written in the form \begin{align*} -\operatorname{div}\left[\frac{|\nabla u|^{p-2}}{|y|^{ap}}\nabla u\right] -μ\,\frac{u^{p-1}}{|y|^{p(a+1)}} = \frac{u^{p^*(a,b)-1}}{|y|^{bp^*(a,b)}} + \frac{u^{p^*(a,c)-1}}{|y|^{cp^*(a,c)}}, \qquad (x,y) \in \mathbb{R}^{N-k}\times\mathbb{R}^k. \end{align*} The existence of a positive, weak solution $u \in \mathcal{D}_a^{1,p}(\mathbb{R}^N\backslash\{|y|=0\})$ is proved with the help of the mou…
This paper deals with the analysis of the internal control with constraint of positive kind of a parabolic PDE with nonlinear diffusion when the time horizon is large enough. The minimal controllability time will be strictly positive. We prove a global steady state constrained controllability result for a quasilinear parabolic with nonlinearity in the diffusion term. Then, under suitable dissipative assumption in the system and local controllability results, we conclude the result to any initial datum and any target trajectory.
We study a two-point free boundary problem in a sector for a quasilinear parabolic equation. The boundary conditions are assumed to be spatially and temporally "self-similar" in a special way. We prove the existence, uniqueness and asymptotic stability of an expanding solution which is self-similar at discrete times. We also study the existence and uniqueness of a shrinking solution which is self-similar at discrete times.
We study the Cauchy problem for fully nonlinear (stochastic) parabolic partial differential equations. We provide both in deterministic and stochastic case the existence of a maximal defined solution for the problem and we provide suitable blow-up criterion. The key idea is the use of the paradifferential operator calculus in order to reduce the fully nonlinear problem into an abstract quasilinear (stochastic) parabolic equation. This allows us to use some recent results on abstract quasilinear (stochastic) evolution equations in Banach spaces. To do so, we analyse the properties of the paradi…
We obtain local boundedness of weak solutions of double phase quasilinear parabolic equations of the form \[u_t-\text{div} \left(|\nabla u|^{p-2}\nabla u+a(x,t)|\nabla u|^{q-2}\nabla u\right)=0,\] where, we have imposed the restrictions $\frac{2N}{N+2}<p<\infty$, $0\leq a(x,t)\leq M$ is measurable and $q < p\frac{N+1}{N-1}$.
We consider mixed local and nonlocal quasilinear parabolic equations of $p$-Laplace type and discuss several regularity properties of weak solutions for such equations. More precisely, we establish local boundeness of weak subsolutions, lower semicontinuity of weak supersolutions as well as upper semicontinuity of weak subsolutions. We also discuss the pointwise behavior of the semicontinuous representatives. Our main results are valid for sign changing solutions. Our approach is purely analytic and is based on energy estimates and the De Giorgi theory.
The global existence of weak solutions to a class of quasilinear parabolic equations with nonlinearities depending on first order terms and integrable data in a moving domain is investigated. The class includes the $p$-Laplace equation as a special case. Weak solutions are shown to be global by obtaining appropriate estimates on the gradient as well as a suitable version of Aubin-Lions lemma in moving domains.
In this paper, we mainly consider the initial boundary problem for a quasilinear parabolic equation \[ u_t-\mathrm{div}\left(|\nabla u|^{p-2}\nabla u\right)=-|u|^{β-1}u+α|u|^{q-2}u, \] where $p>1,β>0$, $q\geq1$ and $α>0$. By using Gagliardo-Nirenberg type inequality, energy method and comparison principle, the phenomena of blowup and extinction are classified completely in the different ranges of reaction exponents.
In this paper we consider a one-dimensional fully parabolic quasilinear Keller-Segel system with critical nonlinear diffusion. We show uniform-in-time boundedness of solutions, which means, that unlike in higher dimensions, there is no critical mass phenomenon in the case of critical diffusion. To this end we utilize estimates from a well-known Lyapunov functional and a recently introduced new Lyapunov-like functional in [3].
This paper is concerned with existence and uniqueness of solutions to two kinds of quasilinear parabolic equations. One is described as the form which includes the porous media and fast diffusion type equations. The other is the Cahn--Hilliard type system. The present paper applies Brézis theory directly to both equations and gives existence results for these two equations even if the domain is unbounded. Moreover, an error estimate is also proved via apriori estimates obtained directly.
In this paper, we obtain necessary conditions and sufficient conditions on the initial data for the local-in-time solvability of the Cauchy problem \[ \partial_t u +(-Δ)^\fracθ{2} u=|x|^{-γ} u^p ,\quad x\in{\bf R}^N, t>0, \qquad u(0)=μ\quad \mbox{in} \quad {\bf R}^N, \] where $N\ge 1$, $0<θ\le2$, $p>1$, $γ>0$ and $μ$ is a nonnegative Radon measure on ${\bf R}^N$. Using these conditions, we attempt to identify the optimal strength of the singularity of $μ$ for the existence of solutions to this problem.
We extend to the parabolic setting some of the ideas originated with Xiao Zhong's proof in \cite{Zhong} of the Hölder regularity of $p-$harmonic functions in the Heisenberg group $\Hn$. Given a number $p\ge 2$, in this paper we establish the $C^{\infty}$ smoothness of weak solutions of quasilinear pde's in $\Hn$ modelled on the equation $$\p_t u= \sum_{i=1}^{2n} X_i \bigg((1+|\nabla_0 u|^2)^{\frac{p-2}{2}} X_i u\bigg).$$
