Vanishing solutions of anisotropic parabolic equations with variable nonlinearity

The aim of this paper is to establish sufficient conditions of the finite time blow-up in solutions of the homogeneous Dirichlet problem for the anisotropic parabolic equations with variable nonlinearity \( u_t = \sum\nolimits_{i = 1}^n {D_i (a_i (x,t)|D_i u|^{p^i (x) - 2} D_i u) + \sum\nolimits_{i = 1}^K {b_i (x,t)|u|^{\sigma _i (x,t) - 2} u} } \) . Two different cases are studied. In the first case ai ≡ ai(x), pi ≡ 2, σi ≡ σi(x, t), and bi(x, t) ≥ 0. We show that in this case every solution corresponding to a “large” initial function blows up in finite time if there exists at least one j for which min σj(x, t) > 2 and either bj > 0, or bj(x, t) ≥ 0 and Σπbj−ρ(t)(x, t) dx 0 depending on σj. In the case of the quasilinear equation with the exponents pi and σi depending only on x, we show that the solutions may blow up if min σi ≥ max pi, bi ≥ 0, and there exists at least one j for which min σj > max pj and bj > 0. We extend these results to a semilinear equation with nonlocal forcing terms and quasilinear equations which combine the absorption (bi ≤ 0) and reaction terms.

We consider stationary anisotropic higher-order partial integrodifferential equations with variable exponents of nonlinearity. It is assumed that the unknown function may nonlinearly enter into the integrand. We establish the conditions of existence and uniqueness of generalized solutions of the Dirichlet problem for these equations.

In this paper, the extension of the generalized finite difference method to the solution of anisotropic elliptical and parabolic partial differential equations is given in the case of considering two space dimensions. The explicit finite difference formulae and the criterion of stability for the anisotropic parabolic equations are given. The stability condition has been expressed in function of the coefficients of the star equation for irregular clouds of nodes. Various anisotropic cases of simulation of cardiac tissue, including elliptic and parabolic equations, have been solved, and the results show the accuracy of the method.

We consider the following nonlinear parabolic equation: ut-div(|∇u|p(x)-2∇u)=f(x,t), where f:Ω×(0,T)→R and the exponent of nonlinearity p(·) are given functions. By using a nonlinear operator theory, we prove the existence and uniqueness of weak solutions under suitable assumptions. We also give a two-dimensional numerical example to illustrate the decay of solutions.

In this article, we consider non-negative solutions of the homogeneous Dirichlet problems of parabolic equations with local or nonlocal nonlinearities, involving variable exponents. We firstly obtain the necessary and sufficient conditions on the existence of blow-up solutions, and also obtain some Fujita-type conditions in bounded domains. Secondly, the blow-up rates are determined, which are described completely by the maximums of the variable exponents. Thirdly, we show that the blow-up occurs only at a single point for the equations with local nonlinearities, and in the whole domain for nonlocal nonlinearities.

The initial-boundary value problems for parabolic equations with variable exponents of nonlinearity and time depended delay are considered. Existence and uniqueness of solutions of these problems are proved.

To characterize fast and slow diffusion components in diffusion-weighted magnetic resonance imaging (DW-MRI) of pediatric Crohn's disease (CD). Overall diffusivity reduction as measured by the apparent diffusion coefficient (ADC) in patients with CD has been previously demonstrated. However, the ADC reduction may be due to changes in either fast or slow diffusion components. In this study we distinguished between the fast and slow diffusion components in the DW-MRI signal decay of pediatric CD. We acquired MRI from 24 patients, including MR enterography (MRE) and DW-MRI with 8 b-values (0-800 s/mm(2)). We characterized fast and slow diffusivity by intravoxel incoherent motion (IVIM) model parameters (f, D*, D), and overall diffusivity by ADC values. We determined which model best described the DW-MRI signal decay. We assessed the influence of the IVIM model parameters on the ADC. We evaluated differences in model parameter values between the enhancing and nonenhancing groups. The IVIM model described the observed data significantly better than the ADC model (P = 0.0088). The ADC was correlated with f (r = 0.67, P = 0.0003), but not with D (r = 0.39, P = 0.062) and D* (r = -0.39, P = 0.057). f values were significantly lower (P < 0.003) and D* values were significantly higher (P = 0.03) in the enhancing segments, while D values were not significantly different between the groups (P = 0.14). For this study population the IVIM model provides a better description of the DW-MRI signal decay than the ADC model. The reduced ADC is related to changes in the fast diffusion rather than to changes in the slow diffusion.

The article investigates removability conditions for singularities of anisotropic parabolic equations and in particular for the anisotropic porous medium equation and it aims in the numerical validation of the analytical results. The preconditions on the strength of the anisotropy are analyzed, and the analytical estimates for the growth behavior of the solutions near the singularities are compared with the observed growth in numerical simulations. Despite classical estimates used in the proof, we find that the analytical estimates are surprisingly close to the numerically observed solution behavior.

The surface Helmholtz integral equation (SHIE) for the acoustic pressure on an object whose surface S vibrates harmonically with a given normal velocity, is known to fail by nonuniqueness at certain “characteristic” frequencies. These frequencies are identical to the eigenfrequencies of the homogeneous Dirichlet problem for a volume of fluid enclosed by the surface S. This paper deals with nondegenerate characteristic frequencies, that is, frequencies at which exactly one independent solution of the interior homogeneous Dirichlet problem exists but no nontrivial solutions of the interior homogeneous Neuman problem exist. The problem is reformulated in terms of operators on the Hilbert space of functions which are square integrable over S. The eigenvectors of the (skew-Hermitian) Green's function operator for the interior Neumann problem provide an orthonormal basis for the space and, hence, the unknown Green's function operator for the exterior Dirichlet problem can be completely specified by its matrix elements with respect to these eigenvectors. The nonuniqueness failure of the SHIE is found to result from the indeterminacy of a single element of this infinite matrix. An expression for the indeterminate matrix element is obtained directly from the SHIE by a perturbation technique yielding a formal solution of the problem.

In this paper, we consider a viscoelastic von Karman equation with damping, delay, and source effects of variable exponent type. Firstly, we show the global existence of solution applying the potential well method. Then, by making use of the perturbed energy method and properties of convex functions, we derive general decay results for the solution under more general conditions of a relaxation function. General decay results of solutions for viscoelastic von Karman equations with variable exponent nonlinearities have not been discussed before. Our results extend and complement many results for von Karman equations in the literature.

CONTENTS § 1. Introduction § 2. A priori estimates for the solutions of linear parabolic equations § 3. A priori estimates for the solutions of quasi-linear parabolic equations § 4. Existence theorems for solutions of the first boundary value problem and Cauchy's problem in the large for quasi-linear parabolic equations § 5. An approximate solution of the boundary value problem and Cauchy's problem for quasi-linear parabolic equations. Generalized solutions of these problems § 6. Theorems on a priori estimates and the solvability of the Dirichlet problem for quasi-linear elliptic equations with many independent variables References

In Sobolev spaces with variable exponent, we consider the problem for a semilinear hyperbolic variational inequality of the third order. We establish conditions for the existence of a solution u of this problem such that u ∈ L ∞((0, T); V 1,0(Ω)), u t ∈ L ∞((0, T); V 1,0(Ω)) ∩ L p(x)(Q T ), and u tt ∈ L ∞((0, T); L 2(Ω)), where V 1,0(Ω) ⊂ H 1(Ω).

Consider an anisotropic parabolic equation with the variable exponents vt=∑i=1n(bi(x,t)vxipi(x)-2vxi)xi+f(v,x,t), where bi(x,t)∈C1(QT¯), pi(x)∈C1(Ω¯), pi(x)&gt;1, bi(x,t)≥0, f(v,x,t)≥0. If {bi(x,t)} is degenerate on Γ2⊂∂Ω, then the second boundary value condition is imposed on the remaining part ∂Ω∖Γ2. The uniqueness of weak solution can be proved without the boundary value condition on Γ2.

We consider a class of degenerate parabolic equaitons on a bounded domain with mixed boundary conditions. These problems arise, for example, in the study of flow through porous media. Under appropriate hypotheses, we establish the existence of a nonegative solution which is obtainable as a monotone limit of solutions of quasilinear parabolic equations. This construction is used establish uniqueness, cinparison, and L1 continuous dependence theorems, as well some results on blow up of solutions in finite time

Consider the anisotropic parabolic equation with the variable exponent ut=∑i=1N(ai(x)|uxi|pi(x)−2uxi)xi,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ {u_{t}}=\\sum_{i=1}^{N} \\bigl(a_{i}(x)|u_{x_{i}}|^{p_{i}(x)-2}u_{x_{i}} \\bigr)_{x _{i}}, $$\\end{document} with a_{i}(x), p_{i}(x)in C^{1}(overline{Omega}), p_{i}(x)>1, a_{i}(x)geq0. If some of {a_{i}(x)} are degenerate on the boundary, a partial boundary value condition is imposed, the stability of weak solutions can be proved based on the partial boundary value condition.