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In this paper, we are concerned with the problem of existence of a capacity solution to the strongly nonlinear degenerate problem, namely, ∂ ∂ t + H () = σ () | ∇ ψ | 2 , div (σ () ∇ ψ) = 0 in Q , where the operator H is of the form H () = ∑ | ν | = 0 ∞ (− 1) | ν | D ν (H ν (t , x , D γ )) , | γ | ≤ | ν |. By using a Schauder's fixed point theorem, we establish the existence of weak solutions to a certain truncated problem of finite order. Then, we demonstrate that the sequence of solutions to this truncated problem converges (up to a subsequence) to a capacity solution to our problem of infinite order. [ABSTRACT FROM AUTHOR]

We consider the homogeneous Dirichlet problem for the parabolic equation ut−div(∣∇u∣p(x,t)−2∇u)=f(x,t)+F(x,t,u,∇u){u}_{t}-{\rm{div}}({| \nabla u| }^{p\left(x,t)-2}\nabla u)=f\left(x,t)+F\left(x,t,u,\nabla u) in the cylinder QT≔Ω×(0,T){Q}_{T}:= \Omega \times \left(0,T), where Ω⊂RN\Omega \subset {{\mathbb{R}}}^{N}, N≥2N\ge 2, is a C2{C}^{2}-smooth or convex bounded domain. It is assumed that p∈C0,1(Q¯T)p\in {C}^{0,1}\left({\overline{Q}}_{T}) is a given function and that the nonlinear source F(x,t,s,ξ)F\left(x,t,s,\xi ) has a proper power growth with respect to ss and ξ\xi . It is shown that if p(x,t)>2(N+1)N+2p\left(x,t)\gt \frac{2\left(N+1)}{N+2}, f∈L2(QT)f\in {L}^{2}\left({Q}_{T}), ∣∇u0∣p(x,0)∈L1(Ω){| \nabla {u}_{0}| }^{p\left(x,0)}\in {L}^{1}\left(\Omega ), then the problem has a solution u∈C0([0,T];L2(Ω))u\in {C}^{0}\left(\left[0,T];\hspace{0.33em}{L}^{2}\left(\Omega )) with ∣∇u∣p(x,t)∈L∞(0,T;L1(Ω)){| \nabla u| }^{p\left(x,t)}\in {L}^{\infty }\left(0,T;\hspace{0.33em}{L}^{1}\left(\Omega )), ut∈L2(QT){u}_{t}\in {L}^{2}\left({Q}_{T}), obtained as the limit of solutions to the regularized problems in the parabolic Hölder space. The solution possesses the following global regularity properties: ∣∇u∣2(p(x,t)−1)+r∈L1(QT),for any 0

This paper aims to study a semilinear transfer equation with a delay term defined over a non-cylindrical domain. We prove the existence and regularity of weak solutions as well as the existence, regularity and finiteness of the fractal dimension of pullback attractors on tempered universes that depend on a non-increasing function. We address the problem of estimating the fractal dimension of pullback attractors, under current techniques, which consists of readjusting the proof presented in [9, Lemma 1.3] and extending it to families of normed spaces parameterized in time, see Theorem 4.11. [ABSTRACT FROM AUTHOR]

In this work, we consider an inverse backward problem for a nonlinear parabolic equation of the Burgers' type with a memory term from final data. To this aim, we first establish the well-posedness of the direct problem. On the basis of the optimal control framework, the existence and necessary condition of the minimizer for the cost functional are established. The global uniqueness and stability of the minimizer are deduced from the necessary condition. Numerical experiments demonstrate the effectiveness of this approach. [ABSTRACT FROM AUTHOR]

In this paper, we prove a local Hamilton type gradient estimate for positive solution of the nonlinear parabolic equation on M × (−∞, ∞) with α ∈ R, where a and b are constants. As application, the Harnack inequalities are derived. [ABSTRACT FROM AUTHOR]

In this paper, we consider a Li-Yau gradient estimate on the positive solution to the following nonlinear parabolic equation ∂/∂t f = Δf + af (ln f)p with Neumann boundary conditions on a compact Riemannian manifold satisfying the integral Ricci curvature assumption, where p ≥ 0 is a real constant. This contrasts Olivé's gradient estimate, which works mainly for the heat equation rather than nonlinear parabolic equations and the result can be regarded as a generalization of the Li-Yau [P. Li, S. T. Yau, On the parabolic kernel of the Schrödinger operator, Acta Math., 156 (1986), 153-201] and Olivé [X. R. Olive, Neumann Li-Yau gradient estimate under integral Ricci curvature bounds, Proc. Amer. Math. Soc., 147 (2019), 411-426] gradient estimates. [ABSTRACT FROM AUTHOR]

Abstract The purpose of this paper is to give a necessary and sufficient condition for the existence and non-existence of global solutions of the following semilinear parabolic equations u t = Δ u + ψ ( t ) f ( u ) , in Ω × ( 0 , ∞ ) , $$ u_{t}=\Delta u+\psi (t)f(u),\quad \text{in }\Omega \times (0,\infty ), $$ under the mixed boundary condition on a bounded domain Ω. In fact, this has remained an open problem for a few decades, even for the case f ( u ) = u p $f(u)=u^{p}$ . As a matter of fact, we prove: there is no global solution for any initial data if and only if ∫ 0 ∞ ψ ( t ) f ( ∥ S ( t ) u 0 ∥ ∞ ) ∥ S ( t ) u 0 ∥ ∞ d t = ∞ for every nonnegative nontrivial initial data u 0 ∈ C 0 ( Ω ) . $$ \begin{aligned} & \text{there is no global solution for any initial data if and only if } \\ & \int _{0}^{\infty}\psi (t) \frac{f (\lVert S(t)u_{0}\rVert _{\infty} )}{\lVert S(t)u_{0}\rVert _{\infty}}\,dt= \infty \\ &\text{for every nonnegative nontrivial initial data } u_{0}\in C_{0}( \Omega ). \end{aligned} $$ Here, ( S ( t ) ) t ≥ 0 $(S(t))_{t\geq 0}$ is the heat semigroup with the mixed boundary condition.

In this paper, we study the existence results for some parabolic equations with degenerate coercivity, singular lower order term depending on the gradient, and positive initial data in $L^1$.

The purpose of this paper is to give a necessary and sufficient condition for the existence and non-existence of global solutions of the following semilinear parabolic equations u t = Δ u + ψ (t) f (u) , in Ω × (0 , ∞) , under the mixed boundary condition on a bounded domain Ω. In fact, this has remained an open problem for a few decades, even for the case f (u) = u p . As a matter of fact, we prove: there is no global solution for any initial data if and only if ∫ 0 ∞ ψ (t) f (∥ S (t) u 0 ∥ ∞) ∥ S (t) u 0 ∥ ∞ d t = ∞ for every nonnegative nontrivial initial data u 0 ∈ C 0 (Ω). Here, (S (t)) t ≥ 0 is the heat semigroup with the mixed boundary condition. [ABSTRACT FROM AUTHOR]

The paper concerns a nonlinear second-order parabolic evolution equation, one of the well-known objects of mathematical physics, which describes the processes of high-temperature thermal conductivity, nonlinear diffusion, filtration of liquid in a porous medium and some other processes in continuum mechanics. A particular case of it is the well-known porous medium equation. Unlike previous studies, we consider the case of several spatial variables. We construct and study solutions that describe disturbances propagating over a zero background with a finite speed, usually called 'diffusion-wave-type solutions'. Such effects are atypical for parabolic equations and appear since the equation degenerates on manifolds where the desired function vanishes. The paper pays special attention to exact solutions of the required type, which can be expressed as either explicit or implicit formulas, as well as a reduction of the partial differential equation to an ordinary differential equation that cannot be integrated in quadratures. In this connection, Cauchy problems for second-order ordinary differential equations arise, inheriting the singularities of the original formulation. We prove the existence of continuously differentiable solutions for them. A new example, an analog of the classic example by S.V. Kovalevskaya for the considered case, is constructed. We also proved a new existence and uniqueness theorem of heat-wave-type solutions in the class of piece-wise analytic functions, generalizing previous ones. During the proof, we transit to the hodograph plane, which allows us to overcome the analytical difficulties. [ABSTRACT FROM AUTHOR]

In this article, we study the long-time dynamical behavior of the solution for a class of semilinear edge-degenerate parabolic equations on manifolds with edge singularities. By introducing a family of potential well and compactness method, we reveal the dependence between the initial data and the long-time dynamical behavior of the solution. Specifically, we give the threshold condition for the initial data, which makes the solution exist globally or blowup in finite-time with subcritical, critical, and supercritical initial energy, respectively. Moreover, we also discussed the long-time behavior of the global solution, the estimate of blowup time, and blowup rate. Our results show that the relationship between the initial data and the long-time behavior of the solution can be revealed in the weighted Sobolev spaces for nonlinear parabolic equations on manifolds with edge singularities.

In this paper, we study the existence results for some parabolic equations with degenerate coercivity, singular lower order term depending on the gradient, and positive initial data in L¹. [ABSTRACT FROM AUTHOR]

A method for the numerical solution of a nonlinear equation describing the diffusion transfer of radiation energy has been developed. According to the method, the second time derivative with a small parameter is introduced into the parabolic equation and an explicit difference scheme is applied. The explicit approximation of the initial equation makes it possible to implement on its basis an algorithm that is effectively adapted to the architecture of high-performance computing systems. In comparison with the original scheme, the new one allows for a larger time integration step and a sufficiently high resolution of the solution structure, providing the second order of accuracy. A heuristic algorithm for choosing the parameters of the three-level difference scheme is proposed. A promising field of application of the method can be problems in plasma physics and astrophysics. [ABSTRACT FROM AUTHOR]

The main target of this paper is to present an efficient method to solve a nonlinear free boundary mathematical model of prostate tumor. This model consists of two parabolics, one elliptic and one ordinary differential equations that are coupled together and describe the growth of a prostate tumor. We start our discussion by using the front fixing method to fix the free domain. Then, after employing a nonclassical finite difference and the collocation methods on this model, their stability and convergence are proved analytically. Finally, some numerical results are considered to show the efficiency of the mentioned methods. [ABSTRACT FROM AUTHOR]

The purpose of this work is to study a nonlinear parabolic equation with singular lower order term of asymptotic type. Existence and uniqueness results are obtained in a suitable Lebesgue and Sobolev spaces with variable exponent. Besides, to verify the effectiveness of the proposed model in noise removal, a numerical experiment is performed and the obtained results are presented. [ABSTRACT FROM AUTHOR]

In this study, we examine a modified heat equation with memory and nonlinear source. The source function is considered under two different conditions, the global Lipschitz and the exponential growth functions. For the first condition, a special weighted Banach space is applied to deduce a desired result without any assumption on sufficiently small time and initial data. For the second condition of exponential growth, we apply the Moser–Trudinger inequality to cope with the source function, and a special time-space norm to deduce the unique existence of a global solution in regard to sufficiently small data. The main objective of this work is to prove the global existence and uniqueness of mild solutions. Besides the solution techniques, our main arguments are also based on the Banach fixed point theorem and linear estimates for the mild solution. The highlight of this study is that it is the first work on the global well-posedness for the mild solution of the fractional heat conduction with memory and nonlinear sources. [ABSTRACT FROM AUTHOR]

We consider the generalized parabolic Cahn–Hilliard equation u t = - Δ Δ u - W ′ (u) + W ′ ′ (u) Δ u - W ′ (u) , for (t , x) ∈ R ~ × R n , where n = 2 or n ≥ 4 , W (·) is the typical double-well potential function and R ~ is given by R ~ = (0 , ∞) , if n = 2 , (- ∞ , 0) , if n ≥ 4. We construct a radial solution u(t, x) possessing an interface. At main order this solution consists of a traveling copy of the steady state ω (| x |) , which satisfies ω ′ ′ (y) - W ′ (ω (y)) = 0 . Its interface is resemble at main order copy of the sphere of the following form | x | = - 2 (n - 3) (n - 1) 2 t 4 , for (t , x) ∈ R ~ × R n , which is a solution to the Willmore flow in Differential Geometry. When n = 1 or 3, the result consists of trivial interface solutions. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the asymptotic behavior of solutions of $ M_1 $ model on quadrant $ (x,t) \in \mathbb{R}^{+} \times \mathbb{R}^{+} $. From this model, combined with damped compressible Euler equations, a more general system is introduced. We show that the solutions to the initial boundary value problem of this system globally exist and tend time-asymptotically to the corresponding nonlinear parabolic equation governed by the related Darcy's law. Compared with previous results on compressible Euler equations with damping obtained by Nishihara and Yang in [26], and Marcati, Mei and Rubino in [17], the better convergence rates are obtained. The approach adopted is based on the technical time-weighted energy estimates together with the Green's function method. [ABSTRACT FROM AUTHOR]

In this work, the solvability for an inverse problem of a nonlinear parabolic equation with nonlocal integral overdetermination supplementary condition is examined. The proof of the existence and uniqueness of the solution of the inverse nonlinear parabolic problem upon the data is established by using the fixed-point technique. In addition, the inverse problem is investigated by using the cubic B-spline collocation technique together with the Tikhonov regularization. The resulting nonlinear system of parabolic equation is approximated using the MATLAB subroutine lsqnonlin. The obtained results demonstrate the accuracy and efficiency of the technique, and the stability of the approximate solutions even in the existence of noisy data. The stability analysis is also conducted for the discretized system of the direct problem.

The paper concerns a nonlinear second-order parabolic evolution equation, one of the well-known objects of mathematical physics, which describes the processes of high-temperature thermal conductivity, nonlinear diffusion, filtration of liquid in a porous medium and some other processes in continuum mechanics. A particular case of it is the well-known porous medium equation. Unlike previous studies, we consider the case of several spatial variables. We construct and study solutions that describe disturbances propagating over a zero background with a finite speed, usually called ‘diffusion-wave-type solutions’. Such effects are atypical for parabolic equations and appear since the equation degenerates on manifolds where the desired function vanishes. The paper pays special attention to exact solutions of the required type, which can be expressed as either explicit or implicit formulas, as well as a reduction of the partial differential equation to an ordinary differential equation that cannot be integrated in quadratures. In this connection, Cauchy problems for second-order ordinary differential equations arise, inheriting the singularities of the original formulation. We prove the existence of continuously differentiable solutions for them. A new example, an analog of the classic example by S.V. Kovalevskaya for the considered case, is constructed. We also proved a new existence and uniqueness theorem of heat-wave-type solutions in the class of piece-wise analytic functions, generalizing previous ones. During the proof, we transit to the hodograph plane, which allows us to overcome the analytical difficulties.

Abstract In this paper, we firstly discuss blow-up phenomena for nonlinear parabolic equations u t = ∇ ⋅ [ ρ ( u ) ∇ u ] + f ( x , t , u ) , in Ω × ( 0 , t ∗ ) , $$ u_{t}=\nabla \cdot \bigl[\rho (u)\nabla u \bigr]+f(x,t,u),\quad \text{in }\Omega \times \bigl(0,t^{*}\bigr), $$ under mixed nonlinear boundary conditions ∂ u ∂ n + θ ( z ) u = h ( z , t , u ) $\frac{\partial u}{\partial n}+\theta (z)u=h(z,t,u)$ on Γ 1 × ( 0 , t ∗ ) $\Gamma _{1}\times (0,t^{*})$ and u = 0 $u=0$ on Γ 2 × ( 0 , t ∗ ) $\Gamma _{2}\times (0,t^{*})$ , where Ω is a bounded domain and Γ 1 $\Gamma _{1}$ and Γ 2 $\Gamma _{2}$ are disjoint subsets of a boundary ∂Ω. Here, f and h are real-valued C 1 $C^{1}$ -functions and ρ is a positive C 1 $C^{1}$ -function. To obtain the blow-up solutions, we introduce the following blow-up conditions: ( C ρ ) : ( 2 + ϵ ) ∫ 0 u ρ ( w ) f ( x , t , w ) d w ≤ u ρ ( u ) f ( x , t , u ) + β 1 u 2 + γ 1 , ( 2 + ϵ ) ∫ 0 u ρ 2 ( w ) h ( z , t , w ) d w ≤ u ρ 2 ( u ) h ( z , t , u ) + β 2 u 2 + γ 2 , $$ (C_{\rho})\,:\, \begin{aligned} &(2+\epsilon ) \int _{0}^{u}\rho (w)f(x,t,w)\,dw\leq u\rho (u)f(x,t,u)+ \beta _{1}u^{2}+\gamma _{1}, \\ &(2+\epsilon ) \int _{0}^{u}\rho ^{2}(w)h(z,t,w)\,dw \leq u\rho ^{2}(u)h(z,t,u)+ \beta _{2}u^{2}+ \gamma _{2}, \end{aligned} $$ for x ∈ Ω $x\in \Omega $ , z ∈ ∂ Ω $z\in \partial \Omega $ , t > 0 $t>0$ , and u ∈ R $u\in \mathbb{R}$ for some constants ϵ, β 1 $\beta _{1}$ , β 2 $\beta _{2}$ , γ 1 $\gamma _{1}$ , and γ 2 $\gamma _{2}$ satisfying ϵ > 0 , β 1 + λ R + 1 λ S β 2 ≤ ρ m 2 λ R 2 ϵ and 0 ≤ β 2 ≤ ρ m 2 λ S 2 ϵ , $$ \epsilon >0,\quad \beta _{1}+\frac{\lambda _{R}+1}{\lambda _{S}}\beta _{2} \leq \frac{\rho _{m}^{2}\lambda _{R}}{2}\epsilon \quad \text{and}\quad 0 \leq \beta _{2}\leq \frac{\rho _{m}^{2}\lambda _{S}}{2}\epsilon , $$ where ρ m : = inf s > 0 ρ ( s ) $\rho _{m}:=\inf_{s>0}\rho (s)$ , λ R $\lambda _{R}$ is the first Robin eigenvalue and λ S $\lambda _{S}$ is the first Steklov eigenvalue. Lastly, we discuss blow-up solutions for nonlinear parabolic systems.

In this paper, we continue to study an initial boundary value problems to a model describing the evolution in time of diffusive phase interfaces in sea ice growth. In a previous paper, the global existence and the large‐time behavior of weak solutions in one space was studied under Dirichlet boundary conditions. Here, we show that the global existence of weak solutions and the large‐time behavior are also studied under Neumann boundary condition. In this paper, we study in space dimension lower than or equal to 3. [ABSTRACT FROM AUTHOR]

In this work, we study the existence of a capacity solution for a nonlocal thermistor problem in Musielak–Orlicz–Sobolev spaces. We get the existence of capacity solution using the approximate techniques and we prove the existence of a weak solution by introducing a sequence of approximate problems converging in a certain sense to a capacity solution. As a consequence, we obtain the existence of a capacity solution of the original problem in Musielak–Orlicz–Sobolev Lebesgue spaces. [ABSTRACT FROM AUTHOR]

Let (\mathbf {M}^n, g) be an n-dimensional complete Riemannian manifold. We prove that for any p>\frac {n}{2}, when k(p,1) is small enough, two certain elliptic-type gradient estimates hold for any positive solutions of the equation \begin{equation*} u_{t}=\Delta u+au\log u \end{equation*} on geodesic balls B(O,r) in \mathbf {M}^n with 0

We developed a technique for the numerical solution of a nonlinear equation describing the diffusion transfer of radiation energy. The method is based on the introduction of the second time derivative with a small parameter into the parabolic equation and an explicit difference scheme. The explicit approximation of the original equation makes it possible to implement an algorithm that is effectively adapted to the architecture of high-performance computing systems. The new scheme provides a second-order resolution of the nonlinearity in time with an acceptable time step. A heuristic algorithm for choosing the parameters of a three-level difference scheme is proposed. Perspective applications of the method are problems in astrophysics, for example, the simulation of a strongly radiating shock wave breakout at the surface of a star at the stage of its evolution known as a supernova explosion. [ABSTRACT FROM AUTHOR]

In this paper we consider nonlinear parabolic equation with power type nonlinearity which such types of problems occur in many mathematical models of applied science. We show that there are solutions under some conditions on initial data which blow up in finite time with positive initial energy. [ABSTRACT FROM AUTHOR]

We study the homogeneous Dirichlet problem for the evolution p(x, t)-Laplacian with the nonlinear source u t - div | ∇ u | p (x , t) - 2 ∇ u = f (x , t , u) , (x , t) ∈ Q = Ω × (0 , T). Here, Ω ⊂ R n is a bounded domain, n ≥ 2 , and p (x , t) is a given function p (·) : Q ↦ (2 n n + 2 , p + ] , p + < ∞ . It is shown that the solution is stable with respect to perturbations of the exponent p(x, t), the nonlinear source f(x, t, u), and the initial datum. We obtain quantitative estimates on the norm of the difference between two solutions in a variable Sobolev space through the norms of perturbations of the exponent p(x, t) and the data u(x, 0), f. Estimates on the rate of convergence of solutions of perturbed problems to the solution of the limit problem are derived. [ABSTRACT FROM AUTHOR]

Abstract A nonlinear parabolic equation from a two-phase problem is considered in this paper. The existence of weak solutions is proved by the standard parabolically regularized method. Different from the related papers, one of diffusion coefficients in the equation, b ( x ) $b(x)$ , is degenerate on the boundary. Then the Dirichlet boundary value condition may be overdetermined. In order to study the stability of weak solution, how to find a suitable partial boundary value condition is the foremost work. Once such a partial boundary value condition is found, the stability of weak solutions will naturally follow.

In this paper, we firstly discuss blow-up phenomena for nonlinear parabolic equations u t = ∇ ⋅ [ ρ (u) ∇ u ] + f (x , t , u) , in Ω × (0 , t ∗) , under mixed nonlinear boundary conditions ∂ u ∂ n + θ (z) u = h (z , t , u) on Γ 1 × (0 , t ∗) and u = 0 on Γ 2 × (0 , t ∗) , where Ω is a bounded domain and Γ 1 and Γ 2 are disjoint subsets of a boundary ∂Ω. Here, f and h are real-valued C 1 -functions and ρ is a positive C 1 -function. To obtain the blow-up solutions, we introduce the following blow-up conditions: (C ρ) : (2 + ϵ) ∫ 0 u ρ (w) f (x , t , w) d w ≤ u ρ (u) f (x , t , u) + β 1 u 2 + γ 1 , (2 + ϵ) ∫ 0 u ρ 2 (w) h (z , t , w) d w ≤ u ρ 2 (u) h (z , t , u) + β 2 u 2 + γ 2 , for x ∈ Ω , z ∈ ∂ Ω , t > 0 , and u ∈ R for some constants ϵ, β 1 , β 2 , γ 1 , and γ 2 satisfying ϵ > 0 , β 1 + λ R + 1 λ S β 2 ≤ ρ m 2 λ R 2 ϵ and 0 ≤ β 2 ≤ ρ m 2 λ S 2 ϵ , where ρ m : = inf s > 0 ρ (s) , λ R is the first Robin eigenvalue and λ S is the first Steklov eigenvalue. Lastly, we discuss blow-up solutions for nonlinear parabolic systems. [ABSTRACT FROM AUTHOR]

This paper is concerned with the initial boundary value problem for a nonlinear parabolic equation with a small parameter. The existence of a boundary layer as the parameter goes to zero is obtained together with the estimation on the thickness of the boundary layer. The main result extends an earlier work of Frid and Shelukhin (1999). [ABSTRACT FROM AUTHOR]

Our research on applied mathematics is in line with the research scope of the journal. The phase-field model is applied to simulate the material and other areas. A phase-field model describing the non-isothermal solidification of an ideal multi-component alloy system is proposed in this paper. The time and space variation of a three-phase-field function and the governing equations of the temperature field are established. The global existence of weak solutions for three-dimensional parabolic differential equations is proved by the Faedo–Galerkin method. The existence of a maximum theorem is also extensively studied. [ABSTRACT FROM AUTHOR]

The paper studies a degenerate nonlinear parabolic equation containing a convective term and a source (reaction) term. It considers the construction of approximate solutions to this equation with a specified law of diffusion wave motion, the existence of these solutions being proved in our previous studies. A stepwise algorithm of the numerical solution with a time-difference scheme is proposed, the second-order difference scheme being used in such problems for the first time. At each step the problem is solved iteratively on the basis of a radial basis function (RBF) collocation method. In order to verify the numerical solution algorithm, two classes of exact generalized traveling wave solutions are proposed, whose construction is reduced to solving a Cauchy problem for second order ordinary differential equations (ODEs) with a singularity at the higher derivative. The theorem of the existence and uniqueness of the analytical solution in the form of a power series is proved for it, and the estimates of the radius of convergence are obtained. The Euler method is used to prove a similar statement concerning the existence of a continuous solution in the non-analytical case. The RBF collocation method is also applied for the approximate solution of the Cauchy problem. The solutions to the Cauchy problem are numerically analyzed, and this has enabled us to reveal and describe some of their properties, including those not previously observed, and to assess the accuracy of the method. [ABSTRACT FROM AUTHOR]

The well-posedness problem of a nonlinear parabolic equation with double variable exponents is studied in this paper. This kind of nonlinear parabolic equation includes the non-Newtonian fluids equation, the polytropic filtration equation and the so-called electro-rheological fluid equation. One of the important characteristics is that there is a diffusion coefficient a(x, t) in the equation. Unlike the usual assumption a (x , t) > a > 0 , the paper only assumes a (x , t) ≥ 0 . If a (x , t) | x ∈ ∂ Ω = 0 , by choosing a(x, t) as a test function, the stability result of positive solutions can be established without the boundary value condition. [ABSTRACT FROM AUTHOR]

Superconvergent property for a nonlinear parabolic equation with 3-step backward differential formula (BDF) Galerkin finite element method (FEM) is studied. The time-discrete system is established to split the error into the temporal error and the spatial error. The scheme of the initial two steps are dealt with a new way to ensure the third order of the temporal error. Gagliardo-Nirenberg interpolation inequality, the linear combination of two adjacent layers and the other skills are utilized to get the spatial error in H 1 -norm of order O (h (h + τ 3 2)) without the ratio between the spatial subdivision parameter h and the temporal step τ . The final superclose result is deduced by the above achievements and the trigonometric inequality. Two numerical examples are provided to support the theoretical analysis. [ABSTRACT FROM AUTHOR]

The Carreau fluid flow in an infinite pipe, caused by a time-dependent pressure gradient, is studied theoretically. The problem is axisymmetric in space and reduces to a nonlinear parabolic equation for the axial velocity. This equation has no analytical solution in the general case of different Carreau numbers. In the present paper, it is proved that the velocity and its gradient, as well as their differences from the similar Newtonian solution, are bounded by explicitly found constants, which depend on the Carreau number and on the other parameters of the viscosity rheological model. These estimates are also discussed for the special case of oscillatory pressure gradient using some numerical examples at different Carreau numbers. [ABSTRACT FROM AUTHOR]

We consider the optimal control problem for solutions of the parabolic equation with the right-hand side of the form ε F (y), where ε > 0 is a small parameter, with the coercive objective functional and bounded control. We use the formula of the optimal controller of the unperturbed problem and substantiate the form of the approximate controller with switching for the original problem. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we propose an efficient B-spline finite element method for a class of fourth order nonlinear differential equations with variable coefficient. For the temporal discretization, we choose the Crank–Nicolson scheme. Boundedness and error estimates are rigorously derived for both semi-discrete and fully discrete schemes. A numerical experiment confirms our theoretical analysis.

A nonlinear parabolic equation from a two-phase problem is considered in this paper. The existence of weak solutions is proved by the standard parabolically regularized method. Different from the related papers, one of diffusion coefficients in the equation, b (x) , is degenerate on the boundary. Then the Dirichlet boundary value condition may be overdetermined. In order to study the stability of weak solution, how to find a suitable partial boundary value condition is the foremost work. Once such a partial boundary value condition is found, the stability of weak solutions will naturally follow. [ABSTRACT FROM AUTHOR]

In this paper, we study blow‐up phenomena of the following p‐Laplace type nonlinear parabolic equations ut=∇·ρ(|∇u|p)|∇u|p−2∇u+f(x,t,u),inΩ×(0,t∗),under nonlinear mixed boundary conditions ρ(|∇u|p)|∇u|p−2∂u∂n+θ(z)ρ(|u|p)|u|p−2u=h(z,t,u),onΓ1×(0,t∗),and u=0 on Γ2 × (0, t∗) such that Γ1∪Γ2=∂Ω, where f and h are real‐valued C1‐functions. To discuss blow‐up solutions, we introduce new conditions: For each x ∈ Ω, z ∈ ∂Ω, t > 0, u > 0, and v > 0, (Dp1):αF(x,t,u)≤uf(x,t,u)+β1up+γ1,αH(z,t,u)≤uh(z,t,u)+β2up+γ2,(Dp2):δvρ(v)≤P(v),for some constants α, β1, β2, γ1, γ2, and δ satisfying α>2,δ>0,β1+λR+1λSβ2≤αδp−1ρmλR,and0≤β2≤αδp−1ρmλS,where ρm:=infw>0ρ(w), P(v)=∫0vρ(w)dw, F(x,t,u)=∫0uf(x,t,w)dw, and H(x,t,u)=∫0uh(x,t,w)dw. Here, λR is the first Robin eigenvalue and λS is the first Steklov eigenvalue for the p‐Laplace operator, respectively. [ABSTRACT FROM AUTHOR]

Abstract In this paper, an efficient finite element scheme is presented for a class of fourth-order nonlinear parabolic problems with variable coefficient. To deal with second-order term in weak formulation, we choose the cubic B-spline function as a trial function. Rigorous error estimates are derived for both semi-discrete and fully-discrete schemes. We provide a numerical example to confirm our theoretical results.

The paper studies a degenerate nonlinear parabolic equation containing a convective term and a source (reaction) term. It considers the construction of approximate solutions to this equation with a specified law of diffusion wave motion, the existence of these solutions being proved in our previous studies. A stepwise algorithm of the numerical solution with a time-difference scheme is proposed, the second-order difference scheme being used in such problems for the first time. At each step the problem is solved iteratively on the basis of a radial basis function (RBF) collocation method. In order to verify the numerical solution algorithm, two classes of exact generalized traveling wave solutions are proposed, whose construction is reduced to solving a Cauchy problem for second order ordinary differential equations (ODEs) with a singularity at the higher derivative. The theorem of the existence and uniqueness of the analytical solution in the form of a power series is proved for it, and the estimates of the radius of convergence are obtained. The Euler method is used to prove a similar statement concerning the existence of a continuous solution in the non-analytical case. The RBF collocation method is also applied for the approximate solution of the Cauchy problem. The solutions to the Cauchy problem are numerically analyzed, and this has enabled us to reveal and describe some of their properties, including those not previously observed, and to assess the accuracy of the method.

In this article we propose a new approach for investigation the local existence of classical solutions of IBVP for a class of nonlinear parabolic equations

A variable stepsize exponential multistep integrator, with contour integral approximation of the operator-valued exponential functions, is proposed for solving semilinear parabolic equations with nonsmooth initial data. By this approach, the exponential k-step method would have k th -order convergence in approximating a mild solution, possibly nonsmooth at the initial time. In consistency with the theoretical analysis, a numerical example shows that the method can achieve high-order convergence in the maximum norm for semilinear parabolic equations with discontinuous initial data. [ABSTRACT FROM AUTHOR]

When the Riemannian metric evolves under the Ricci flow, we investigate parabolic gradient estimates (Li–Yau's type and J. Li's type) for positive solutions to the nonlinear parabolic equation (Δ - ∂ t) u = (p + 1) | ∇ u | 2 u + q u on the underlying manifold. Based on these gradient estimates, we derive associated Harnack inequalities, respectively. [ABSTRACT FROM AUTHOR]

Our research on applied mathematics is in line with the research scope of the journal. The phase-field model is applied to simulate the material and other areas. A phase-field model describing the non-isothermal solidification of an ideal multi-component alloy system is proposed in this paper. The time and space variation of a three-phase-field function and the governing equations of the temperature field are established. The global existence of weak solutions for three-dimensional parabolic differential equations is proved by the Faedo–Galerkin method. The existence of a maximum theorem is also extensively studied.