Journal für die reine und angewandte Mathematik (Crelles Journal), ISSN 0075-4102, 03/2016, Volume 2016, Issue 712, pp. 51 - 80

We study a quasilinear parabolic equation of forward-backward type, under assumptions on the nonlinearity which hold for a wide class of mathematical models,...

MATHEMATICS | PSEUDOPARABOLIC REGULARIZATION | HEAT | SHEAR-FLOW | DIRECTION | CONVERGENCE | PERONA-MALIK EQUATION | MODEL | DIFFUSION EQUATION

MATHEMATICS | PSEUDOPARABOLIC REGULARIZATION | HEAT | SHEAR-FLOW | DIRECTION | CONVERGENCE | PERONA-MALIK EQUATION | MODEL | DIFFUSION EQUATION

Journal Article

2.
Smoothing and decay estimates for nonlinear diffusion equations

: equations of porous medium type

2006, Oxford lecture series in mathematics and its applications, ISBN 0199202974, Volume 33, xiii, 234

This book is concerned with the quantitative aspects of the theory of nonlinear diffusion equations; equations which can be seen as nonlinear variations of the...

Burgers equation | applied mathematics | Delayed regularity | Singular parabolicity | Smoothing | Nonlinearities of power type | Asymptotics | Extinction in finite time | Time decay | Decay rates | Classical heat equation

Burgers equation | applied mathematics | Delayed regularity | Singular parabolicity | Smoothing | Nonlinearities of power type | Asymptotics | Extinction in finite time | Time decay | Decay rates | Classical heat equation

Book

Nonlinear Analysis, ISSN 0362-546X, 2010, Volume 73, Issue 11, pp. 3507 - 3512

The present paper is concerned with the blow up rate of a solution to the following Cauchy problem u t = Δ u m + u p , ( x , t ) ∈ R N × ( 0 , T ) , with the...

Porous medium equation | Blow up rate | Cauchy problem | MATHEMATICS | MATHEMATICS, APPLIED | SEMILINEAR HEAT-EQUATIONS | Mathematical analysis | Images | Proving | Oscillations | Nonlinearity | Estimates

Porous medium equation | Blow up rate | Cauchy problem | MATHEMATICS | MATHEMATICS, APPLIED | SEMILINEAR HEAT-EQUATIONS | Mathematical analysis | Images | Proving | Oscillations | Nonlinearity | Estimates

Journal Article

Journal of Mathematical Physics, ISSN 0022-2488, 12/2012, Volume 53, Issue 12, p. 123515

Preliminary group classification became a prominent tool in the symmetry analysis of differential equations due to the paper by Ibragimov, Torrisi, and Valenti...

HEAT | EIKONAL EQUATION | SYMMETRIES | PARTIAL-DIFFERENTIAL-EQUATIONS | DIFFUSION-EQUATIONS | ALGEBRA | POWER NONLINEARITIES | LIE | CONSERVATION-LAWS | PHYSICS, MATHEMATICAL | TRANSFORMATIONS | Algebra | Wave equations | Classification | Group theory | Differential equations | Nonlinearity | Transformations | Symmetry

HEAT | EIKONAL EQUATION | SYMMETRIES | PARTIAL-DIFFERENTIAL-EQUATIONS | DIFFUSION-EQUATIONS | ALGEBRA | POWER NONLINEARITIES | LIE | CONSERVATION-LAWS | PHYSICS, MATHEMATICAL | TRANSFORMATIONS | Algebra | Wave equations | Classification | Group theory | Differential equations | Nonlinearity | Transformations | Symmetry

Journal Article

5.
Pseudoparabolic regularization of forward-backward parabolic equations: A logarithmic nonlinearity

Analysis and PDE, ISSN 2157-5045, 2013, Volume 6, Issue 7, pp. 1719 - 1754

We study the initial-boundary value problem {u(t) = Delta phi(u) + epsilon Delta[psi(u)](t) in Q := Omega x (0, T], phi(u) + epsilon[psi(u)](t) = 0 in partial...

Bounded radon measures | Entropy inequalities | Pseudoparabolic regularization | Forward-backward parabolic equations | forward-backward parabolic equations | bounded radon measures | MATHEMATICS | HEAT | MATHEMATICS, APPLIED | SHEAR-FLOW | pseudoparabolic regularization | DIFFUSION | MODEL | entropy inequalities

Bounded radon measures | Entropy inequalities | Pseudoparabolic regularization | Forward-backward parabolic equations | forward-backward parabolic equations | bounded radon measures | MATHEMATICS | HEAT | MATHEMATICS, APPLIED | SHEAR-FLOW | pseudoparabolic regularization | DIFFUSION | MODEL | entropy inequalities

Journal Article

Analysis, ISSN 0174-4747, 03/2018, Volume 38, Issue 1, pp. 21 - 36

The initial value problem for an inhomogeneous nonlinear heat equation with a pure power nonlinearity is investigated. In the radial energy space, global and...

Nonlinear heat equation | global well-posedness | 35K55 | blow-up

Nonlinear heat equation | global well-posedness | 35K55 | blow-up

Journal Article

7.
Full Text
Blow-up and non-extinction for a nonlocal parabolic equation with logarithmic nonlinearity

Boundary Value Problems, ISSN 1687-2762, 12/2018, Volume 2018, Issue 1, pp. 1 - 11

This paper is devoted to studying a nonlocal parabolic equation with logarithmic nonlinearity ulog|u|−⨏Ωulog|u|dx $u\log |u|-\fint_{\Omega } u\log |u|\,dx$ in...

Nonlocal parabolic equation | Ordinary Differential Equations | Analysis | Difference and Functional Equations | Approximations and Expansions | Mathematics, general | Mathematics | Partial Differential Equations | Blow-up | Non-extinction | MATHEMATICS | SEMILINEAR HEAT-EQUATION | MATHEMATICS, APPLIED | SINGULARITY | P-LAPLACE EQUATION | NEUMANN BOUNDARY-CONDITIONS | LIOUVILLE-TYPE THEOREMS | SUPERLINEAR PROBLEMS | CRITICAL EXPONENTS | Extinction | Nonlinearity

Nonlocal parabolic equation | Ordinary Differential Equations | Analysis | Difference and Functional Equations | Approximations and Expansions | Mathematics, general | Mathematics | Partial Differential Equations | Blow-up | Non-extinction | MATHEMATICS | SEMILINEAR HEAT-EQUATION | MATHEMATICS, APPLIED | SINGULARITY | P-LAPLACE EQUATION | NEUMANN BOUNDARY-CONDITIONS | LIOUVILLE-TYPE THEOREMS | SUPERLINEAR PROBLEMS | CRITICAL EXPONENTS | Extinction | Nonlinearity

Journal Article

8.
Full Text
Elliptic Equations with Multi-Singular Inverse-Square Potentials and Critical Nonlinearity

Communications in Partial Differential Equations, ISSN 0360-5302, 03/2006, Volume 31, Issue 3, pp. 469 - 495

This article deals with a class of nonlinear elliptic equations involving a critical power-nonlinearity as well as a potential featuring multiple inverse...

Critical sobolev exponent | Hardy inequality | Multi-singular potentials | Concentration compactness Principle | concentration compactness Principle | EXISTENCE | MATHEMATICS | MATHEMATICS, APPLIED | critical sobolev exponent | EXPONENTS | INEQUALITIES | multi-singular potentials | HEAT-EQUATION

Critical sobolev exponent | Hardy inequality | Multi-singular potentials | Concentration compactness Principle | concentration compactness Principle | EXISTENCE | MATHEMATICS | MATHEMATICS, APPLIED | critical sobolev exponent | EXPONENTS | INEQUALITIES | multi-singular potentials | HEAT-EQUATION

Journal Article

Nonlinear Analysis, ISSN 0362-546X, 2008, Volume 68, Issue 3, pp. 461 - 484

This paper studies the Cauchy problem for the nonlinear fractional power dissipative equation u t + ( − △ ) α u = F ( u ) for initial data in the Lebesgue...

Space–time estimates | Fractional power dissipative equation | Besov spaces | Cauchy problem | Well-posedness | Space-time estimates | EXISTENCE | MATHEMATICS, APPLIED | well-posedness | SEMILINEAR PARABOLIC EQUATIONS | EVOLUTION-EQUATIONS | space-time estimates | MATHEMATICS | fractional power dissipative equation | HEAT-EQUATION | QUASI-GEOSTROPHIC EQUATION | TRIEBEL-LIZORKIN SPACES | Operators | Mathematical analysis | Triangles | Dissipation | Images | Nonlinearity | Mathematics - Analysis of PDEs

Space–time estimates | Fractional power dissipative equation | Besov spaces | Cauchy problem | Well-posedness | Space-time estimates | EXISTENCE | MATHEMATICS, APPLIED | well-posedness | SEMILINEAR PARABOLIC EQUATIONS | EVOLUTION-EQUATIONS | space-time estimates | MATHEMATICS | fractional power dissipative equation | HEAT-EQUATION | QUASI-GEOSTROPHIC EQUATION | TRIEBEL-LIZORKIN SPACES | Operators | Mathematical analysis | Triangles | Dissipation | Images | Nonlinearity | Mathematics - Analysis of PDEs

Journal Article

SIAM Journal on Applied Mathematics, ISSN 0036-1399, 1/2012, Volume 72, Issue 3, pp. 935 - 958

Finite time singularity formation in a fourth order nonlinear parabolic partial differential equation (PDE) is analyzed. The PDE is a variant of a ubiquitous...

Microelectromechanical systems | Geometry | Electric potential | Capacitors | Far fields | Eigenvalues | Touchdown | Boundary conditions | Mathematics | Symmetry | Singularity formation | Biharmonic equations | Self-similar solutions | MATHEMATICS, APPLIED | EIGENVALUE PROBLEMS | HEAT-EQUATIONS | BEHAVIOR | STABILITY | ELLIPTIC EQUATION | PARTIAL-DIFFERENTIAL-EQUATIONS | ELECTROSTATIC MEMS | DIELECTRIC-PROPERTIES | touchdown | singularity formation | biharmonic equations | BLOW-UP | self-similar solutions | EXTINCTION | Partial differential equations | Singularities | Mathematical analysis | Nonlinearity | Mathematical models | Quenching | Self-similarity | Mathematics - Analysis of PDEs

Microelectromechanical systems | Geometry | Electric potential | Capacitors | Far fields | Eigenvalues | Touchdown | Boundary conditions | Mathematics | Symmetry | Singularity formation | Biharmonic equations | Self-similar solutions | MATHEMATICS, APPLIED | EIGENVALUE PROBLEMS | HEAT-EQUATIONS | BEHAVIOR | STABILITY | ELLIPTIC EQUATION | PARTIAL-DIFFERENTIAL-EQUATIONS | ELECTROSTATIC MEMS | DIELECTRIC-PROPERTIES | touchdown | singularity formation | biharmonic equations | BLOW-UP | self-similar solutions | EXTINCTION | Partial differential equations | Singularities | Mathematical analysis | Nonlinearity | Mathematical models | Quenching | Self-similarity | Mathematics - Analysis of PDEs

Journal Article

SIAM Journal on Applied Mathematics, ISSN 0036-1399, 1/2010, Volume 70, Issue 7/8, pp. 3319 - 3341

We consider singular solutions of the L²-critical biharmonic nonlinear Schrödinger equation. We prove that the blowup rate is bounded by a quartic-root, the...

Standing waves | Conservation laws | Statistical variance | Heat equation | Applied mathematics | Ground state | Nonlinearity | Mathematical inequalities | Spectral methods | Blowup | NLS | Nonlinear Schrödinger | Self-similar solutions | Biharmonic | High-order dispersion | MATHEMATICS, APPLIED | high-order dispersion | DISPERSION | nonlinear Schrodinger | blowup | GLOBAL WELL-POSEDNESS | CRITICAL POWER NONLINEARITY | biharmonic | MEDIA | 4TH-ORDER | BLOW-UP SOLUTIONS | CRITICAL DIMENSION | self-similar solutions | Studies | Nonlinear equations | Schrodinger equation | Asymptotic methods

Standing waves | Conservation laws | Statistical variance | Heat equation | Applied mathematics | Ground state | Nonlinearity | Mathematical inequalities | Spectral methods | Blowup | NLS | Nonlinear Schrödinger | Self-similar solutions | Biharmonic | High-order dispersion | MATHEMATICS, APPLIED | high-order dispersion | DISPERSION | nonlinear Schrodinger | blowup | GLOBAL WELL-POSEDNESS | CRITICAL POWER NONLINEARITY | biharmonic | MEDIA | 4TH-ORDER | BLOW-UP SOLUTIONS | CRITICAL DIMENSION | self-similar solutions | Studies | Nonlinear equations | Schrodinger equation | Asymptotic methods

Journal Article

Heat and Mass Transfer, ISSN 0947-7411, 3/2017, Volume 53, Issue 3, pp. 1037 - 1049

The performance characteristics and temperature field of conducting–convecting–radiating annular fin are investigated. The nonlinear variation of thermal...

Engineering | Thermodynamics | Industrial Chemistry/Chemical Engineering | Engineering Thermodynamics, Heat and Mass Transfer | DEPENDENT THERMAL-CONDUCTIVITY | PROFILES | MECHANICS | THERMODYNAMICS | SIMPLEX SEARCH METHOD | PERFORMANCE | OPTIMAL DIMENSIONS | OPTIMIZATION | GENERATION | HEAT-TRANSFER COEFFICIENT | PARAMETERS | CIRCULAR FINS | Mechanical engineering | Differential equations | Electric properties

Engineering | Thermodynamics | Industrial Chemistry/Chemical Engineering | Engineering Thermodynamics, Heat and Mass Transfer | DEPENDENT THERMAL-CONDUCTIVITY | PROFILES | MECHANICS | THERMODYNAMICS | SIMPLEX SEARCH METHOD | PERFORMANCE | OPTIMAL DIMENSIONS | OPTIMIZATION | GENERATION | HEAT-TRANSFER COEFFICIENT | PARAMETERS | CIRCULAR FINS | Mechanical engineering | Differential equations | Electric properties

Journal Article

13.
Full Text
On the Heat Equation with Nonlinearity and Singular Anisotropic Potential on the Boundary

Potential Analysis, ISSN 0926-2601, 3/2017, Volume 46, Issue 3, pp. 589 - 608

This paper concerns with the heat equation in the half-space ℝ + n $\mathbb {R}_{+}^{n}$ with nonlinearity and singular potential on the boundary ∂ ℝ + n...

35K20 | 35B07 | 35C06 | 35B06 | Nonlinear boundary conditions | Heat equation | 35A01 | Probability Theory and Stochastic Processes | 35K05 | Mathematics | Geometry | 42B35 | Potential Theory | Functional Analysis | Lorentz spaces | Singular potentials | Self-similarity | Symmetry | EXISTENCE | INEQUALITY | PARABOLIC PROBLEMS | MATHEMATICS | ATTRACTORS | CRITICAL GROWTH | NAVIER-STOKES EQUATION | ELLIPTIC-EQUATIONS | INITIAL DATA | WEAK SOLUTIONS | SCHRODINGER-OPERATORS | Anisotropy | Analysis

35K20 | 35B07 | 35C06 | 35B06 | Nonlinear boundary conditions | Heat equation | 35A01 | Probability Theory and Stochastic Processes | 35K05 | Mathematics | Geometry | 42B35 | Potential Theory | Functional Analysis | Lorentz spaces | Singular potentials | Self-similarity | Symmetry | EXISTENCE | INEQUALITY | PARABOLIC PROBLEMS | MATHEMATICS | ATTRACTORS | CRITICAL GROWTH | NAVIER-STOKES EQUATION | ELLIPTIC-EQUATIONS | INITIAL DATA | WEAK SOLUTIONS | SCHRODINGER-OPERATORS | Anisotropy | Analysis

Journal Article

Transactions of the American Mathematical Society, ISSN 0002-9947, 03/2006, Volume 358, Issue 3, pp. 1165 - 1185

We study large time asymptotics of small solutions to the Cauchy problem for nonlinear damped wave equations with a critical nonlinearity \begin{equation*}...

Preliminary estimates | Damping | Heat equation | Wave equations | Nonlinearity | Fourier transformations | Mathematics | Cauchy problem | Damped wave equation | Large time asymptotics | MATHEMATICS | CRITICAL EXPONENT | GLOBAL EXISTENCE | DECAY | BEHAVIOR | damped wave equation | CAUCHY-PROBLEM | large time asymptotics

Preliminary estimates | Damping | Heat equation | Wave equations | Nonlinearity | Fourier transformations | Mathematics | Cauchy problem | Damped wave equation | Large time asymptotics | MATHEMATICS | CRITICAL EXPONENT | GLOBAL EXISTENCE | DECAY | BEHAVIOR | damped wave equation | CAUCHY-PROBLEM | large time asymptotics

Journal Article

Journal of the Acoustical Society of America, ISSN 0001-4966, 09/2011, Volume 130, Issue 3, pp. 1125 - 1132

Fractional derivatives are well suited to describe wave propagation in complex media. When introduced in classical wave equations, they allow a modeling of...

ACOUSTICS | MARINE-SEDIMENTS | FREQUENCY POWER-LAW | ULTRASONIC-ATTENUATION | TIME-DOMAIN | DISPERSION | AUDIOLOGY & SPEECH-LANGUAGE PATHOLOGY | AMPLITUDE SOUND BEAMS | MEDIA | DIFFUSION | HEAT-CONDUCTION | PROPAGATION | Viscosity | Motion | Fourier Analysis | Time Factors | Stress, Mechanical | Acoustics | Elastic Modulus | Entropy | Sound | Nonlinear Dynamics | Thermal Conductivity | Mathematical analysis | Wave equations | Attenuation | Nonlinearity | Mathematical models | Derivatives | Dispersions

ACOUSTICS | MARINE-SEDIMENTS | FREQUENCY POWER-LAW | ULTRASONIC-ATTENUATION | TIME-DOMAIN | DISPERSION | AUDIOLOGY & SPEECH-LANGUAGE PATHOLOGY | AMPLITUDE SOUND BEAMS | MEDIA | DIFFUSION | HEAT-CONDUCTION | PROPAGATION | Viscosity | Motion | Fourier Analysis | Time Factors | Stress, Mechanical | Acoustics | Elastic Modulus | Entropy | Sound | Nonlinear Dynamics | Thermal Conductivity | Mathematical analysis | Wave equations | Attenuation | Nonlinearity | Mathematical models | Derivatives | Dispersions

Journal Article

Acta Mechanica, ISSN 0001-5970, 6/2016, Volume 227, Issue 6, pp. 1727 - 1742

In this paper, oscillators with asymmetric and symmetric quadratic nonlinearity are compared. Both oscillators are modeled as ordinary second-order...

Engineering | Vibration, Dynamical Systems, Control | Engineering Thermodynamics, Heat and Mass Transfer | Theoretical and Applied Mechanics | Continuum Mechanics and Mechanics of Materials | Structural Mechanics | Classical Continuum Physics | HOMOCLINIC BIFURCATION | SINGLE-DEGREE | MECHANICS | DRIVEN HELMHOLTZ | RESONANCE | ROUTES | CHAOTIC MOTION | VALID ASYMPTOTIC SOLUTION | SIMULATION | Comparative analysis | Models | Differential equations | Nonlinear systems | Oscillators | Approximation | Asymmetry | Mathematical analysis | Nonlinearity | Mathematical models | Symmetry

Engineering | Vibration, Dynamical Systems, Control | Engineering Thermodynamics, Heat and Mass Transfer | Theoretical and Applied Mechanics | Continuum Mechanics and Mechanics of Materials | Structural Mechanics | Classical Continuum Physics | HOMOCLINIC BIFURCATION | SINGLE-DEGREE | MECHANICS | DRIVEN HELMHOLTZ | RESONANCE | ROUTES | CHAOTIC MOTION | VALID ASYMPTOTIC SOLUTION | SIMULATION | Comparative analysis | Models | Differential equations | Nonlinear systems | Oscillators | Approximation | Asymmetry | Mathematical analysis | Nonlinearity | Mathematical models | Symmetry

Journal Article

COMPUTATIONAL MATHEMATICS AND MATHEMATICAL PHYSICS, ISSN 0965-5425, 06/2019, Volume 59, Issue 6, pp. 1015 - 1029

The paper is devoted to constructing approximate heat wave solutions propagating along the cold front at a finite speed for a nonlinear (quasi-linear) heat...

nonlinear heat conduction equation | numerical solution | MATHEMATICS, APPLIED | boundary element method | heat wave | PARABOLIC EQUATION | PHYSICS, MATHEMATICAL | special series | Methods | Algorithms | Conductive heat transfer | Conduction | Cold fronts | Wave propagation | Heat exchange | Conduction heating | Nonlinearity | Software | Nonlinear programming | Power series | Boundary element method

nonlinear heat conduction equation | numerical solution | MATHEMATICS, APPLIED | boundary element method | heat wave | PARABOLIC EQUATION | PHYSICS, MATHEMATICAL | special series | Methods | Algorithms | Conductive heat transfer | Conduction | Cold fronts | Wave propagation | Heat exchange | Conduction heating | Nonlinearity | Software | Nonlinear programming | Power series | Boundary element method

Journal Article

Mathematical Problems in Engineering, ISSN 1024-123X, 2/2015, Volume 2015, pp. 1 - 7

We propose a power series extender method to obtain approximate solutions of nonlinear differential equations. In order to assess the benefits of this...

MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | ENGINEERING, MULTIDISCIPLINARY | HOMOTOPY | HEAT-TRANSFER | Series | Research | Differential equations, Nonlinear | Mathematical research | Approximation | Perturbation methods | Mathematical analysis | Differential equations | Nonlinearity | Taylor series | Mathematical models | Power series | Convergence

MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | ENGINEERING, MULTIDISCIPLINARY | HOMOTOPY | HEAT-TRANSFER | Series | Research | Differential equations, Nonlinear | Mathematical research | Approximation | Perturbation methods | Mathematical analysis | Differential equations | Nonlinearity | Taylor series | Mathematical models | Power series | Convergence

Journal Article

Nonlinear Analysis, ISSN 0362-546X, 2009, Volume 71, Issue 5, pp. 2236 - 2256

We study the behavior of solutions of the Cauchy problem for a semilinear parabolic equation with a singular power nonlinearity. It is known for a...

Singularity | Stationary solution | Critical exponent | Semilinear heat equation | Cauchy problem | Convergence | MATHEMATICS, APPLIED | CAUCHY-PROBLEM | ELLIPTIC EQUATION | SUPERCRITICAL NONLINEARITY | DIFFUSION EQUATION | GROW-UP RATE | MATHEMATICS | HEAT-EQUATION

Singularity | Stationary solution | Critical exponent | Semilinear heat equation | Cauchy problem | Convergence | MATHEMATICS, APPLIED | CAUCHY-PROBLEM | ELLIPTIC EQUATION | SUPERCRITICAL NONLINEARITY | DIFFUSION EQUATION | GROW-UP RATE | MATHEMATICS | HEAT-EQUATION

Journal Article

20.
Full Text
Periodicity and ergodicity for abstract evolution equations with critical nonlinearities

Advances in Difference Equations, ISSN 1687-1839, 12/2015, Volume 2015, Issue 1, pp. 1 - 13

We study the existence of pseudo almost periodic mild solutions for the abstract evolution equation u ′ ( t ) = A u ( t ) + f ( t , u ( t ) )...

critical nonlinearities | 35B15 | 35K05 | pseudo almost periodic solutions | Mathematics | 58J35 | Ordinary Differential Equations | Functional Analysis | Analysis | Difference and Functional Equations | Mathematics, general | growth conditions | Partial Differential Equations | EXISTENCE | MATHEMATICS | MATHEMATICS, APPLIED | CAUCHY-PROBLEMS | BANACH-SPACES | DIFFERENTIAL-EQUATIONS | PARABOLIC PROBLEMS | DELAY | Ergodic theory | Periodic functions | Differential equations | Difference equations | Mathematical analysis | Texts | Nonlinearity | Evolution | Ergodic processes | Heat equations

critical nonlinearities | 35B15 | 35K05 | pseudo almost periodic solutions | Mathematics | 58J35 | Ordinary Differential Equations | Functional Analysis | Analysis | Difference and Functional Equations | Mathematics, general | growth conditions | Partial Differential Equations | EXISTENCE | MATHEMATICS | MATHEMATICS, APPLIED | CAUCHY-PROBLEMS | BANACH-SPACES | DIFFERENTIAL-EQUATIONS | PARABOLIC PROBLEMS | DELAY | Ergodic theory | Periodic functions | Differential equations | Difference equations | Mathematical analysis | Texts | Nonlinearity | Evolution | Ergodic processes | Heat equations

Journal Article

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