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In this paper, we give a short proof of the Lorentz estimates for gradients of very weak solutions to the linear parabolic equations with the Muckenhoupt class A q -weights u t − div ( A ( x , t ) ∇ u ) = div ( F ) , in a bounded domain Ω × ( 0 , T ) ⊂ R N + 1 , where A has a small mean oscillation, and Ω is a Lipchistz domain with a small Lipschitz constant.

Recently, it was shown that the solution of the Helmholtz equation can be approximated by a series over the solutions of iterative parabolic equations (IPEs). An expansion of the fundamental solution of the Helmholtz equation over solutions of IPEs is considered. It is shown that the resulting Taylor-like series can be easily transformed into a Padé-type approximation. In practical propagation problems such iterative Padé approximations exhibit improved wide-angle capabilities and faster convergence to the solution of the Helmholtz equation in comparison to Taylor-like expansion over IPEs solutions. A Gaussian smoothing of the expansion terms gives insight into the derivation of initial conditions consistent for IPEs, which can be used for point source simulation. A correct point source model consistent with the wide-angle one-way propagation equations is important in many practical applications of the parabolic equations theory.

In this paper we develop a potential theory for strongly degenerate parabolic operators of the form L : = ∇ X ⋅ ( A ( X , Y , t ) ∇ X ) + X ⋅ ∇ Y − ∂ t , in unbounded domains of the form Ω = { ( X , Y , t ) = ( x , x m , y , y m , t ) ∈ R m − 1 × R × R m − 1 × R × R | x m > ψ ( x , y , y m , t ) } , where ψ is assumed to satisfy a uniform Lipschitz condition adapted to the dilation structure and the (non-Euclidean) Lie group underlying the operator L . Concerning A = A ( X , Y , t ) we assume that A is bounded, measurable, symmetric and uniformly elliptic (as a matrix in R m ). Beyond the solvability of the Dirichlet problem and other fundamental properties our results include scale and translation invariant boundary comparison principles, boundary Harnack inequalities and doubling properties of associated parabolic measures. All of our estimates are translation- and scale-invariant with constants only depending on the constants defining the boundedness and ellipticity of A and the Lipschitz constant of ψ. Our results represent a version, for operators of Kolmogorov type with bounded, measurable coefficients, of the by now classical results of Fabes and Safonov, and several others, concerning boundary estimates for uniformly parabolic equations in (time-dependent) Lipschitz type domains.

We study the property of extinction in a finite time for nonnegative solutions of 1 q ∂ ∂ t ( u q ) − ∇ ( | ∇ u | p − 2 ∇ u ) + a ( x ) u λ = 0 for the Dirichlet Boundary Conditions when q > λ > 0, p ⩾ 1 + q, p ⩾ 2, a ( x ) ⩾ 0 and Ω a bounded domain of R N ( N ⩾ 1). We prove some necessary and sufficient conditions. The threshold is for power functions when p > 1 + q while finite time extinction occurs for very flat potentials a ( x ) when p = 1 + q.

In the Cauchy problem L1u≡Lu+(b,∇u)+cu-ut=0,(x,t)∈D,u(x,0)=u0(x),x∈RN, for nondivergent parabolic equation with growing lower-order term in the half-space D=RN×[0,∞), N⩾3, we prove sufficient conditions for exponential stabilization rate of solution as t→+∞ uniformly with respect to x on any compact K in RN with any bounded and continuous in RN initial function u0(x).

We introduce the theory of random tensors, which naturally extends the method of random averaging operators in our earlier work (Deng et al. in: Invariant Gibbs measures and global strong solutions for 4009 nonlinear Schrodinger equations in dimension two, arXiv:1910.08492 ), to study the propagation of randomness under nonlinear dispersive equations. By applying this theory we establish almost-sure local well-posedness for semilinear Schrodinger equations in the full subcritical range relative to the probabilistic scaling (Theorem 1.1). The solution we construct has an explicit expansion in terms of multilinear Gaussians with adapted random tensor coefficients. As a byproduct we also obtain new results concerning regular data and long-time solutions, in particular Theorem 1.6, which provides long-time control for random homogeneous data, demonstrating the highly nontrivial fact that the first energy cascade happens at a much later time than in the deterministic setting. In the random setting, the probabilistic scaling is the natural scaling for dispersive equations, and is different from the natural scaling for parabolic equations. Our theory of random tensors can be viewed as the dispersive counterpart of the existing parabolic theories (regularity structures, para-controlled calculus and renormalization group techniques).

We obtain Calder\'on-Zygmund type estimates for parabolic equations with Orlicz growth, where nonlinearities involved in the equations may be discontinuous for the space and time variables. In addition, we consider parabolic systems with the Uhlenbeck structure., Comment: 19pages

This paper devotes to the existence of an N N -dimensional inertial manifold for a class of singularly, i.e. A ( t ) A(t) may degenerate to 0 0 at some time t t , non-autonomous parabolic equations ∂ t u + A ( t ) u = F ( t , u ) + g ( x , t ) , t > τ ; u | t = τ = u τ ( x ) , x ∈ Ω , \begin{equation*} \partial _{t}u+A(t)u=F(t,u)+g(x,t),\;t>\tau ;\; \; u|_{t=\tau }=u_{\tau }(x),\;x\in \Omega , \end{equation*} where A ( t ) ≥ 0 A(t)\geq 0 for any t ≥ τ t\geq \tau , and Ω ⊂ R d \Omega \subset \mathbb {R}^{d} is a bounded domain with smooth boundary. Since the operator A ( t ) A(t) may degenerate, a compatibility condition for the operator A ( t ) A(t) and the nonlinear term F ( t , u ) F(t,u) was proposed to construct the inertial manifolds.

We consider the first mixed problem for a certain class of anisotropic parabolic equations of the form $$ {\left(\beta \left(x,u\right)\right)}_t^{\prime }-\operatorname{div}\kern0.5em a\left(t,x,u,\nabla u\right)-b\left(t,x,u,\nabla u\right)=u $$ where μ is a measure and the coefficients contain nonpower nonlinearities in the cylindrical domain DT = (0, T)×Ω, where Ω ⊂ ℝn is a bounded domain. We prove the existence of a renormalized solution of the problem for gt = 0 and a function β(x, r), which increases with respect to r and satisfies the Carath´eodory condition.

We continue our study of bounded solutions of the semilinear parabolic equation u t = u x x + f ( u ) on the real line, where f is a locally Lipschitz function on R . Assuming that the initial value u 0 = u ( ⋅ , 0 ) of the solution has finite limits θ ± as x → ± ∞ , our goal is to describe the asymptotic behavior of u ( x , t ) as t → ∞ . In a prior work, we showed that if the two limits are distinct, then the solution is quasiconvergent, that is, all its locally uniform limit profiles as t → ∞ are steady states. It is known that this result is not valid in general if the limits are equal: θ ± = θ 0 . In the present paper, we have a closer look at the equal-limits case. Under minor non-degeneracy assumptions on the nonlinearity, we show that the solution is quasiconvergent if either f ( θ 0 ) ≠ 0 , or f ( θ 0 ) = 0 and θ 0 is a stable equilibrium of the equation ξ ˙ = f ( ξ ) . If f ( θ 0 ) = 0 and θ 0 is an unstable equilibrium of the equation ξ ˙ = f ( ξ ) , we also prove some quasiconvergence theorem making (necessarily) additional assumptions on u 0 . A major ingredient of our proofs of the quasiconvergence theorems—and a result of independent interest—is the classification of entire solutions of a certain type as steady states and heteroclinic connections between two disjoint sets of steady states.

We study regularity properties of variational solutions to a class of Cauchy–Dirichlet problems of the form { ∂ t ⁡ u - div x ⁡ ( D ξ ⁢ f ⁢ ( D ⁢ u ) ) = 0 in ⁢ Ω T , u = u 0 on ⁢ ∂ 𝒫 ⁡ Ω T . \left\{\begin{aligned} \displaystyle\partial_{t}u-\operatorname{div}_{x}(D_{% \xi}f(Du))&\displaystyle=0&&\displaystyle\phantom{}\text{in }\Omega_{T},\\ \displaystyle u&\displaystyle=u_{0}&&\displaystyle\phantom{}\text{on }\partial% _{\mathcal{P}}\Omega_{T}.\end{aligned}\right. We do not impose any growth conditions from above on f : ℝ n → ℝ {f\colon\mathbb{R}^{n}\to\mathbb{R}} , but only require it to be convex and coercive. The domain Ω ⊂ ℝ n {\Omega\subset\mathbb{R}^{n}} is mainly supposed to be bounded and convex, and for the time-independent boundary datum u 0 : Ω ¯ → ℝ {u_{0}\colon\overline{\Omega}\to\mathbb{R}} we only require continuity. These requirements are weaker than a one-sided bounded slope condition. We prove global continuity of the unique variational solution u : Ω T → ℝ {u\colon\Omega_{T}\to\mathbb{R}} . If the boundary datum is Lipschitz continuous, we obtain global Hölder continuity of the solution.

In the presence of vacuum, the physical entropy for polytropic gases behaves singularly and it is thus a challenge to study its dynamics. It is shown in this paper that the boundedness of the entropy can be propagated up to any finite time provided that the initial vacuum presents only at far fields with sufficiently slow decay of the initial density. More precisely, for the Cauchy problem of the one dimensional heat conductive compressible Navier-Stokes equations, the global well-posedness of strong solutions and uniform boundedness of the corresponding entropy are established, as long as the initial density vanishes only at far fields with a rate no more than $O(\frac{1}{x^2})$. The main tools of proving the uniform boundedness of the entropy are some singularly weighted energy estimates carefully designed for the heat conductive compressible Navier-Stokes equations and an elaborate De Giorgi type iteration technique for some classes of degenerate parabolic equations. The De Giorgi type iterations are carried out to different equations in establishing the lower and upper bounds of the entropy.

This paper is devoted to discussing the blow-up problem of the positive solution of the following degenerate parabolic equations: ( r ( u ) ) t = div ( ∣ ∇ u ∣ p ∇ u ) + f ( x , t , u , ∣ ∇ u ∣ 2 ) , ( x , t ) ∈ D × ( 0 , T ∗ ) , ∂ u ∂ ν + σ u = 0 , ( x , t ) ∈ ∂ D × ( 0 , T ∗ ) , u ( x , 0 ) = u 0 ( x ) , x ∈ D ¯ . \left\{\begin{array}{ll}{(r\left(u))}_{t}={\rm{div}}(| \nabla u{| }^{p}\nabla u)+f\left(x,t,u,| \nabla u\hspace{-0.25em}{| }^{2}),& \left(x,t)\in D\times \left(0,{T}^{\ast }),\\ \frac{\partial u}{\partial \nu }+\sigma u=0,& \left(x,t)\in \partial D\times \left(0,{T}^{\ast }),\\ u\left(x,0)={u}_{0}\left(x),& x\in \overline{D}.\end{array}\right. Here p > 0 p\gt 0 , the spatial region D ⊂ R n ( n ≥ 2 ) D\subset {{\mathbb{R}}}^{n}\hspace{0.33em}\left(n\ge 2) is bounded, and its boundary ∂ D \partial D is smooth. We give the conditions that cause the positive solution of this degenerate parabolic problem to blow up. At the same time, for the positive blow-up solution of this problem, we also obtain an upper bound of the blow-up time and an upper estimate of the blow-up rate. We mainly carry out our research by means of maximum principles and first-order differential inequality technique.

For a second-order parabolic equation, we consider a one-sided boundary-value problem with impulsive conditions with respect to the time variable. The coefficients of equation and the boundary conditions have power singularities of any order in time and space variables on a certain set of points. We also establish conditions for the existence and uniqueness of solution to the posed problem in Holder spaces with power weights.

The classical Cox–Ingersoll–Ross (CIR) bond-pricing model is based on the evolution space-time dependent partial differential equation (PDE) which represents the standard European interest rate derivatives. In general, such class of evolution partial differential equations (PDEs) has generally been resolved by classical methods of PDEs and by ansatz-based techniques which have been previously applied in a similar context. The author here shows the application of an invariant approach, a systematic method based on deductive group-theoretical analysis. The invariant technique reduces the scalar linear space-time dependent parabolic PDE to one of the four classical Lie canonical forms. This method leads us to exactly solve the scalar linear space-time dependent parabolic PDE representing the CIR model. It was found that CIR PDE is transformed into the first canonical form, which is the heat equation. Under the proper choice of emerging parameters of the model, the CIR equation is also reduced to the second Lie canonical form. The equivalence transformations which map the CIR PDE into the different canonical forms are deduced. With the use of these equivalence transformations, the invariant solutions of the underlying model are found by using some well-known results of the heat equation and the second Lie canonical form. Furthermore, the Cauchy initial-value model of the CIR problem along with the terminal condition is discussed and closed-form solutions are deduced. Finally, the conservation laws associated with the CIR equation are derived by using the general conservation theorem.

The aim of this paper is to study the solvability of solutions of non-local problems with integral conditions in spatial variables for high-order linear parabolic equations in the classes of regular solutions (which have all the squared derivatives generalized by S. L. Sobolev that are included in the corresponding equation) . Previously, similar problems were studied for high-order parabolic equations, either in the one-dimensional case, or when certain conditions of smallness on the coefficients are met equations. In this paper, we present new results on the solvability of non-local problems with integral spatial variables for high-order parabolic equations a) in the multidimensional case with respect to spatial variables; b) in the absence of smallness conditions. The research method is based on the transition from a problem with non-local integral conditions to a problem with classical homogeneous conditions of the first or second kind on the side boundary for a loaded integro-differential equation. At the end of the paper, some generalizations of the obtained results will be described.

Many models for the movement of particles and individuals are based on the diffusion equation, which, in turn, can be derived from an uncorrelated random walk or a position-jump process. In those models, individuals have a location but no well-defined velocity. An alternative, and sometimes more accurate, model is based on a correlated random walk or a velocity-jump process, where individuals have a well defined location and velocity. The latter approach leads to hyperbolic equations for the density of individuals, rather than parabolic equations that result from the diffusion process. Almost all previous work on correlated random walks considers a homogeneous landscape, whereas diffusion models for uncorrelated walks have been extended to spatially varying environments. In this work, we first derive the equations for a correlated random walk in a one-dimensional spatially varying environment with either smooth variation or piecewise constant variation. Then we show how to derive the so-called parabolic limit from the resulting hyperbolic equations. We develop homogenization theory for the hyperbolic equations, and show that taking the parabolic limit and homogenization are commuting actions. We illustrate our results with two examples from ecology: the persistence and spread of a population in a patchy heterogeneous landscape.

For abstract nonlinear difference schemes with operators acting in finite-dimensional Banach spaces, a stability criterion is stated and proved; namely, for a consistent finite-difference approximation to a well-posed differential problem, the solution of the difference scheme converges if and only if the scheme is unconditionally stable. In a sense, this criterion generalizes Lax’s equivalence theorem to nonlinear differential problems. The results obtained are used to study the stability of difference schemes that approximate quasilinear parabolic equations with nonlinearities of unbounded growth.

The purpose of this work is to study a class of periodic parabolic equations having a critical growth nonlinearity with respect to the gradient and bounded Radon measure. By the main of the sub- and super-solution method, we employ some new technics to prove the existence of a nonnegative weak periodic solution to the studied problems.

We prove the existence of pullback attractors in various spaces for the following non-autonomous quasilinear degenerate parabolic equations involving weighted p-Laplacian operators on $\mathbb {R}^{N}$ $$ \frac{\partial u}{\partial t}-\text{div}(\sigma(x)|\nabla u|^{p-2}\nabla u)+\lambda|u|^{p-2}u+f(u)=g(x,t), $$ under a new condition concerning a variable non-negative diffusivity σ(x), an arbitrary polynomial growth order of the non-linearity f, and an exponential growth of the external force. To overcome the essential difficulty arising due to the unboundedness of the domain, the results are proved by combining the tail estimates method and the asymptotic a priori estimate method.

In this paper we address the following parabolic equation @@ on a smooth metric measure space with Bakry–Emery curvature bounded from below for F being a differentiable function defined on $\mathbb {R}$ . Our motivation is originally inspired by gradient estimates of Allen–Cahn and Fisher-KKP equations (Bǎilesteanu, M., Ann. Glob. Anal. Geom. 51, 367–378, 2017; Cao et al., Pac. J. Math. 290, 273–300, 2017). We show new gradient estimates for these equations. As their applications, we obtain Liouville type theorems for positive or bounded solutions to the above equation when either F = cu(1 − u) (the Fisher-KKP equation) or; F = −u3 + u (the Allen–Cahn equation); or $F=au\log u$ (the equation involving gradient Ricci solitons).

We study convergence of the evolving finite element semidiscretization of a parabolic partial differential equation on an evolving bulk domain. The boundary of the domain evolves with a given velocity, which is then extended to the bulk by solving a Poisson equation. The numerical solution to the parabolic equation depends on the numerical evolution of the bulk, which yields the time-dependent mesh for the finite element method. The stability analysis works with the matrix–vector formulation of the semidiscretization only and does not require geometric arguments, which are then required in the proof of consistency estimates. We present various numerical experiments that illustrate the proven convergence rates.

We study the Cauchy problem for nonlinear degenerate parabolic equations with almost periodic initial data. Existence and uniqueness (in the Besicovitch space) of entropy solutions are established. It is demonstrated that the entropy solution remains to be spatially almost periodic and that its spectrum (more precisely, the additive group generated by the spectrum) does not increase in the time variable. Under a precise nonlinearity-diffusivity condition on the input data we establish the long time decay property in the Besicovitch norm. For the proof we use reduction to the periodic case and ergodic methods.

The objective of this paper is two-fold. The first objective is to complete the former work of Bensoussan and Frehse [2] . One big limitation of this paper was the fact that they are systems of PDE. on a bounded domain. One can expect solutions to be bounded, since one looks for smooth solutions. This is a very important property for the development of the method. It is true also that solutions which exist in a bounded domain may fail to exist on R n , because of the lack of bounds. We give conditions so that the results of [2] can be extended to R n . The second objective is to consider the BSDE (Backward stochastic differential equations) version of the system of PDE. This is the objective of a more recent work of Xing and Zitkovic [8] . They consider systems of BSDE with quadratic growth, which is a well-known open problem in the BSDE literature. Since the BSDE are Markovian, the problem is equivalent to the analytic one. However, because of this motivation the analytic problem is in R n and not on a bounded domain. Xing and Zitkovic developed a probabilistic approach. The connection between the analytic problem and the BSDE is not apparent. Our objective is to show that the analytic approach can be completely translated into a probabilistic one. Nevertheless probabilistic concepts are also useful, after their conversion into the analytic framework. This is in particular true for the uniqueness result.

In this paper we establish a stability result for the nonnegative local weak solutions to $$\begin{aligned} u_t= \text {div} \big (|Dw|^{p-2}Dw\big ) , \quad p>1 \end{aligned}$$ where $$w= \frac{u^\gamma -1}{\gamma }$$ and $$\gamma = \frac{m+p-2}{p-1}$$ , as $$|\gamma |\rightarrow 0$$ .

This article extends the semidiscrete maximal $L^p$-regularity results in Li (2019, Analyticity, maximal regularity and maximum-norm stability of semi-discrete finite element solutions of parabolic equations in nonconvex polyhedra. Math. Comp., 88, 1--44) to multistep fully discrete finite element methods for parabolic equations with more general diffusion coefficients in $W^{1,d+\beta }$, where $d$ is the dimension of space and $\beta>0$. The maximal angles of $R$-boundedness are characterized for the analytic semigroup $e^{zA_h}$ and the resolvent operator $z(z-A_h)^{-1}$, respectively, associated to an elliptic finite element operator $A_h$. Maximal $L^p$-regularity, an optimal $\ell ^p(L^q)$ error estimate and an $\ell ^p(W^{1,q})$ estimate are established for fully discrete finite element methods with multistep backward differentiation formulae.

In this paper, we examine the solvability of a new class of nonclassical direct problems for multidimensional loaded parabolic equations with Cauchy data. We obtain sufficient conditions for the solvability of the problem; the proof is based on the method of weak approximation. By an example, we demonstrate the application of the theorem proved to the study of inverse problems for multidimensional parabolic equations with Cauchy data.

In this paper, we consider a loaded parabolic equation of a special form in an unbounded domain with Cauchy data. The equation is one-dimensional and its right-hand side depends on the unknown function u(t, x) and traces of this function and its derivatives by the spatial variable at a finite number of different points of space. Such equations appear after the reduction of some identification problems for coefficients of one-dimensional parabolic equations with Cauchy data to auxiliary direct problems. We obtain sufficient conditions of the global solvability and sufficient conditions of the solvability of the problem considered in a small time interval. We search for solutions in the class of sufficiently smooth bounded functions. We examine the uniqueness of the classical solution found and prove the corresponding sufficient conditions. We also obtain an a priori estimate of a solution that guarantees the continuous dependence of the solution on the right-hand side of the equation and the initial conditions.

International audience; In this work we determine the second-order coefficient in a parabolic equation from the knowledge of a single final data. Under assumptions on the concentration of eigenvalues of the associated elliptic operator, and the initial state, we show the uniqueness of solution, and we derive a Lipschitz stability estimate for the inversion when the final time is large enough. The Lipschitz stability constant grows exponentially with respect to the final time, which makes the inversion ill-posed. The proof of the stability estimate is based on a spectral decomposition of the solution to the parabolic equation in terms of the eigenfunctions of the associated elliptic operator, and an ad hoc method to solve a nonlinear stationary transport equation that is itself of interest.

In this paper, we present results of the development of certain parallel numerical methods for solving three-dimensional evolutionary and stationary problems of diffusion and heat transfer. We present a detailed description of a special, explicit iteration scheme for parabolic equations and discuss a multigrid technology used for solving elliptic equations and implicit schemes for parabolic equations.

We study a new mathematical model of locally nonequilibrium processes of heat, mass, and momentum transfer taking into account the relaxation phenomena based on hyperbolic and parabolic equations. We also propose a method aimed at getting a priori Schauder-type estimates. The unique solvability of the problem with inner-boundary nonlocal conditions is established.

We demonstrate a measure theoretical approach to the local regularity of weak supersolutions to elliptic and parabolic equations in divergence form. In the first part, we show that weak supersolutions become lower semicontinuous after redefinition on a set of measure zero. The proof relies on a general principle, i.e. the De Giorgi type lemma, which offers a unified approach for a wide class of elliptic and parabolic equations, including an anisotropic elliptic equation, the parabolic p-Laplace equation, and the porous medium equation. In the second part, we shall show that for parabolic problems the lower semicontinuous representative at an instant can be recovered pointwise from the “ess lim inf” of past times. We also show that it can be recovered by the limit of certain integral averages of past times. The proof hinges on the expansion of positivity for weak supersolutions. Our results are structural properties of partial differential equations, independent of any kind of comparison principle.

In this paper, we are concerned with the existence, uniqueness and long-time behavior of the solutions for a parabolic equation with nonlocal diffusion even if the reaction term is not Lipschitz-continuous at $ 0 $ and grows superlinearly or exponentially at $ +\infty $. First, we give a special sub-supersolution pair for some parabolic equations with nonlocal diffusion and establish the method of sub-supersolution. Second, using the sub-supersolution method, we prove the existence, uniqueness and long-time behavior of positive solutions. Finally, some one-dimensional numerical experiments are presented.

We establish a Schn$\ddot{\text{u}}$rer's convergence result and then apply it to obtain the existence of solutions on the second boundary value problem for a family of special Lagrangian equations

A notion of parabolic C-subsolutions is introduced for parabolic equations, extending the theory of C-subsolutions recently developed by B. Guan and more specifically G. Sz\'ekelyhidi for elliptic equations. The resulting parabolic theory provides a convenient unified approach for the study of many geometric flows.

This paper deals with the existence and non-xistence of the solutions to the Cauchy problem of a class of quasilinear parabolic equations with a gradient term. We establish Fujita-type blowup theorems and determine the critical Fujita exponent in terms of spatial dimension, the asymptotic behavior of the coefficients of the gradient term at infinity, the exponents of spatial positions in the coefficients of the time-derivative term, and the source term. In particular, we classify the critical case as the blowup case.

In this paper, we consider the long-time behavior for a class of semi-linear degenerate parabolic equations with the nonlinearity f f satisfying the polynomial growth of arbitrary p − 1 p-1 order. We establish some new estimates, i.e., asymptotic higher-order integrability for the difference of the solutions near the initial time. As an application, we obtain the ( L 2 ( Ω ) , L p ( Ω ) ) \left({L}^{2}\left(\Omega ),{L}^{p}\left(\Omega )) -global attractors immediately; moreover, such an attractor can attract every bounded subset of L 2 ( Ω ) {L}^{2}\left(\Omega ) with the L p + δ {L}^{p+\delta } -norm for any δ ∈ [ 0 , + ∞ ) \delta \in \left[0,+\infty ) .

In this paper, we prove Hamilton type gradient estimates for positive solutions to a general type of nonlinear parabolic equation concerning $ V $-Laplacian: \begin{document}$ (\Delta_{V}-q(x, t)-\partial_{t})u(x, t) = A(u(x, t)) $\end{document} on complete Riemannian manifold (with fixed metric). When $ V = 0 $ and the metric evolves under the geometric flow, we also derive some Hamilton type gradient estimates. Finally, as applications, we obtain some Liouville type theorems of some specific parabolic equations.

In this article, we study the initial boundary value problem of coupled semi-linear degenerate parabolic equations with a singular potential term on manifolds with corner singularities. Firstly, we introduce the corner type weighted p-Sobolev spaces and the weighted corner type Sobolev inequality, the Poincare inequality, and the Hardy inequality. Then, by using the potential well method and the inequality mentioned above, we obtain an existence theorem of global solutions with exponential decay and show the blow-up in finite time of solutions for both cases with low initial energy and critical initial energy. Significantly, the relation between the above two phenomena is derived as a sharp condition. Moreover, we show that the global existence also holds for the case of a potential well family.

In this paper, the finite-time bounded control problem for coupled parabolic PDE-ODE systems subject to time-varying boundary disturbances and to time-invariant boundary disturbances is considered. First, the concept of finite-time boundedness is extended to coupled parabolic PDE-ODE systems. A Neumann boundary feedback controller is then designed in terms of the state variables. By applying the Lyapunov-like functional method, sufficient conditions which ensure the finite-time boundedness of closed-loop systems in the presence of time-varying boundary disturbances and time-invariant boundary disturbances are provided, respectively. Finally, the issues regarding the finite-time boundedness of coupled parabolic PDE-ODE systems are converted into the feasibility of linear matrix inequalities (LMIs), and the effectiveness of the proposed results is validated with two numerical simulations.

In this paper, we study non-divergence form elliptic and parabolic equations with singular coefficients. Weighted and mixed-norm Lp-estimates and solvability are established under suitable partially weighted BMO conditions on the coefficients. When the coefficients are constants, the operators are reduced to extensional operators which arise in the study of fractional heat equations and fractional Laplace equations. Our results are new even in this setting and in the unmixed norm case. For the proof, we explore and utilize the special structures of the equations to show both interior and boundary Lipschitz estimates for solutions and for higher-order derivatives of solutions to homogeneous equations. We then employ the perturbation method by using the Fefferman–Stein sharp function theorem, the Hardy–Littlewood maximal function theorem, as well as a weighted Hardy’s inequality.