On global attractor for nonlinear parabolic equations of m-Laplacian type

Existence and some regularity results of global attractor in Lq, q?1, for m-Laplacian type quasilinear parabolic equation with a perturbation like a(x)(α|u|u−β|u|u)+f(x) with α>β?0, a(x)?0 are proved. For the proofs Moser's technique is used extensively.     

Quadrature-difference methods for solving linear and nonlinear singular integro-differential equations

Here we propose and justify quadrature-difference methods for solving different kinds (linear, nonlinear and multidimensional) of periodic singular integro-differential equations.     

Uniqueness and non-uniqueness of bounded solutions to singular nonlinear parabolic equations

We investigate well-posedness of initial-boundary value problems for a class of nonlinear parabolic equations with variable density. At some part of the boundary, called singular boundary, the density can either vanish or diverge or not need to have a limit. We provide simple conditions for uniqueness or non-uniqueness of bounded solutions, depending on the behaviour of the density near the singular boundary.     

On the theory of nonlinear singular integral equations of Cauchy type

The paper concerns the investigation of two classes of nonlinear singular integral equations with Cauchy kernel important in the applications. There are given uniqueness and existence theorems, the last by a novel application of Schauder's fixed point theorem to this type of equations.     

Existence and nonexistence of solutions for singular quadratic quasilinear equations

We study both existence and nonexistence of nonnegative solutions for nonlinear elliptic problems with singular lower order terms that have natural growth with respect to the gradient, whose model is

     

On the maximum principle for linear parabolic equations

We prove extensions of our previous estimates for linear elliptic equations with inhomogeneous terms in L ^p spaces, p ≤ n to linear parabolic equations with inhomogeneous terms in L ^p , p ≤ n + 1. As with the elliptic case, our results depend on restrictions on parabolicity determined by certain subcones of the positive cone . They also extend the maximum principle of Krylov for the case p = n + 1, corresponding to the usual parabolicity.     

On positivity and stability of linear time-varying Volterra equations

Linear time-varying Volterra integro-differential equations of non-convolution type are considered. Positivity is characterized and a sufficient condition for exponential asymptotic stability is presented.

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The second author thanks the Alexander von Humboldt Foundation for their support.     

Regularity of boundary points for linear equations of parabolic type

The paper considers a second-order linear parabolic equation whose coefficients satisfy a Dini condition. It is proven that the conditions for regularity of the boundary points for such an equation and for the heat-conduction equation coincide.Translated from Matematicheskie Zametki, Vol. 20, No. 5, pp. 717–723, November, 1976.     

Solution of generalized bisymmetry type equations without surjectivity assumptions

Summary. The solution of the rectangular m ×n m \times n generalized bisymmetry equation¶¶F(G₁(x₁₁,...,x_1n),..., G_m(x_m1,...,x_mn))     =     G(F₁(x₁₁,..., x_m1),...,  F_n(x_1n,...,x_mn) ) F\bigl(G_1(x_{11},\dots,x_{1n}),\dots,\ G_m(x_{m1},\dots,x_{mn})\bigr) \quad = \quad G\bigl(F_1(x_{11},\dots, x_{m1}),\dots, \ F_n(x_{1n},\dots,x_{mn}) \bigr) (A)¶is presented assuming that the functions F, G_j, G and F_i (j = 1, ... , m , i = 1, ... , n , m S 2, n S 2) are real valued and defined on the Cartesian product of real intervals, and they are continuous and strictly monotonic in each real variable. Equation (A) is reduced to some special bisymmetry type equations by using induction methods. No surjectivity assumptions are made.     

Symmetrization for linear and nonlinear fractional parabolic equations of porous medium type

We establish symmetrization results for the solutions of the linear fractional diffusion equation _∂tu+(−Δ)^σ/2u=f${\partial }_{t}u+{(-\mathrm{\Delta })}^{\sigma /2}u=f$ and its elliptic counterpart hv+(−Δ)^σ/2v=f$hv+{(-\mathrm{\Delta })}^{\sigma /2}v=f$, h>0$h>0$, using the concept of comparison of concentrations. The results extend to the nonlinear version, _∂tu+(−Δ)^σ/2A(u)=f${\partial }_{t}u+{(-\mathrm{\Delta })}^{\sigma /2}A(u)=f$, but only when the nondecreasing function A:_R+→_R+$A:{\mathbb{R}}_{+}\to {\mathbb{R}}_{+}$ is concave. In the elliptic case, complete symmetrization results are proved for B(v)+(−Δ)^σ/2v=f$B(v)+{(-\mathrm{\Delta })}^{\sigma /2}v=f$ when B(v)$B(v)$ is a convex nonnegative function for v>0$v>0$ with B(0)=0$B(0)=0$, and partial results hold when B is concave. Remarkable counterexamples are constructed for the parabolic equation when A is convex, resp. for the elliptic equation when B   is concave. Such counterexamples do not exist in the standard diffusion case σ=2$\sigma =2$.     

On a class of Volterra nonlinear equations of parabolic type

In this paper, we show the backward uniqueness in time of solution for a class of Volterra nonlinear equations of parabolic type. We also give reasonable physical interpretation for our conclusion.     

Doubly nonlinear parabolic equations with singular lower order term

In this paper we are concerned with positive solutions of the doubly nonlinear parabolic equation u_t=div(u^m−1|∇u|^p−2∇u)+Vu^m+p−2 in a cylinder $\text{Ω×(0,T)}$, with initial condition u(·,0)=u₀(·)⩾0 and vanishing on the parabolic boundary $\text{∂Ω×(0,T)}$. Here $\text{Ω⊂}\text{R}{}^{\mathrm{N}}$ (resp. $\text{H}{}^{\mathrm{n}}$) is a bounded domain with smooth boundary, $\text{V∈L}{}_{\mathrm{<mtext>loc</mtext>}}{}^1\text{(Ω)}$, $\text{m∈}\text{R}$, 1<p<N and m+p−2>0. The critical exponents $\text{q}{}^1$ are found and the nonexistence results are proved for $\text{q}{}^1\text{⩽m+p<3}$.     

Global Lipschitz regularizing effects for linear and nonlinear parabolic equations

In this paper we prove global bounds on the spatial gradient of viscosity solutions to second order linear and nonlinear parabolic equations in (0,T)×R^N$(0,T)\times {\mathbb{R}}^N$. Our assumptions include the case that the coefficients be both unbounded and with very mild local regularity (possibly weaker than the Dini continuity), the estimates only depending on the parabolicity constant and on the modulus of continuity of coefficients (but not on their L^∞$L^{\infty }$-norm). Our proof provides the analytic counterpart to the probabilistic proof used in Priola and Wang (2006) [35] (J. Funct. Anal. 2006) to get this type of gradient estimates in the linear case. We actually extend such estimates to the case of possibly unbounded data and solutions as well as to the case of nonlinear operators including Bellman–Isaacs equations. We investigate both the classical regularizing effect (at time t>0$t>0$) and the possible conservation of Lipschitz regularity from t=0$t=0$, and similarly we prove global Hölder estimates under weaker assumptions on the coefficients. The estimates we prove for unbounded data and solutions seem to be new even in the classical case of linear equations with bounded and Hölder continuous coefficients. Applications to Liouville type theorems are also given in the paper. Finally, we compare in an appendix the analytic and the probabilistic approach discussing the analogy between the doubling variables method of viscosity solutions and the probabilistic coupling method.     

On the nonexistence of entire solutions of certain type of nonlinear differential equations

By utilizing Nevanlinna's value distribution theory of meromorphic functions, it is shown that the following type of nonlinear differential equations:

fⁿ(z)+P_n−3(f)=p₁e^α₁z+p₂e^α₂z