We consider a parabolic-type PDE with a diffusion given by a fractional Laplacian operator and with a quadratic nonlinearity of the 'gradient' of the solution, convoluted with a singular term b. Our first result is the well-posedness for this problem: We show existence and uniqueness of a (local in time) mild solution. The main result is about blow-up of said solution, and in particular we find sufficient conditions on the initial datum and on the term b to ensure blow-up of the solution in finite time.
A second-order PDE is derived from Euler's equaitons under certain assumptions. It is shown that this PDE admits shock and rarefaction waves, and that a single point gradient blow-up admits a unique similarity extension after blow-up that settles uniqueness/entropy issues for such equations.
In this article, we study the local existence of solutions for a wave equation with a nonlocal in time nonlinearity. Moreover, a blow-up results are proved under some conditions on the dimensional space, the initial data and the nonlinear forcing term.
In this paper, we deal with existence, uniqueness and exact rate of boundary behavior of blow-up solutions for a class of logistic type quasilinear problem in a smooth bounded domain involving the $p$-Laplacian operator, where the nonlinearity can have a singular behavior. In the proof of the existence of solution, we have used the sub and super solution method in conjunction with variational techniques and comparison principles. Related to the rate on boundary and uniqueness, we combine a comparison principle proved in the present paper together with the our result of existence of solution.
Various shock and rarefaction-type similarity solutions of the third-order nonlinear dispersion equation in 1D are constructed. Blow-up of some solutions are proved by different techniques.
We investigate the $p-$Laplace heat equation $u_t-Δ_p u=ζ(t)f(u)$ on a bounded smooth domain $Ω\subset\mathbb{R}^N$. Using differential inequalities arguments, we prove blow-up results under suitable conditions on $ζ, f$, and the initial data $u_0$. We also give an upper bound for the blow-up time in each case.
In this paper, we study the blow-up of solutions to the semilinear Moore-Gibson-Thompson (MGT) equation with nonlinearity of derivative type $|u_t|^p$ in the conservative case. We apply an iteration method in order to study both the subcritical case and the critical case. Hence, we obtain a blow-up result for the semilinear MGT equation (under suitable assumptions for initial data) when the exponent $p$ for the nonlinear term satisfies $1<p\leqslant (n+1)/(n-1)$ for $n\geqslant2$ and $p>1$ for $n=1$. In particular, we find the same blow-up range for $p$ as in the corresponding semilinear wave …
In this paper, we study the nonlinear Schrödinger equation with focusing point nonlinearity in dimension one. First, we establish a scattering criterion for the equation based on Kenig-Merle's compactness-rigidity argument. Then we prove the energy scattering below and above the mass-energy threshold. We also describe the dynamics of solutions with data at the ground state threshold. Finally, we prove a blow-up criteria for the equation with initial data with arbitrarily large energy.
This paper is devoted to an integrable two-component Camassa-Holm system with cubic nonlinearity, which includes the cubic Camassa-Holm equation (also called the Fokas-Olver-Rosenau-Qiao equation) as a special case. The one peaked solitons (peakons) and two peakon solutions are described in an explicit formula. Then, the local well-posedness for the Cauchy problem of the system is studied. Moreover, we target at the precise blow-up scenario for strong solutions to the system, and establish a new blow-up result with respect to the initial data.
We study blow-up behavior of solutions for the Cauchy problem of the semilinear wave equation with time-dependent damping. When the damping is effective, and the nonlinearity is subcritical, we show the blow-up rates and the sharp lifespan estimates of solutions. Upper estimates are proved by an ODE argument, and lower estimates are given by a method of scaling variables.
An improvement of [18] on the blow-up region and the lifespan estimate of a weakly coupled system of wave equations with damping and mass in the scale-invariant case and with time-derivative nonlinearity is obtained in this article. Indeed, thanks to a better understanding of the dynamics of the solutions, we give here a better characterization of the blow-up region. Furthermore, the techniques used in this article may be extended to other systems and interestingly they simplify the proof of the blow-up result in [3] which is concerned with the single wave equation in the same context as in th…
In this note, we prove a blow-up result for the semilinear damped wave equation in a compact Lie group with power nonlinearity $|u|^p$ for any $p>1$, under suitable integral sign assumptions for the initial data, by using an iteration argument. A byproduct of this method is the upper bound estimate for the lifespan of a local in time solution. As a preliminary result, a local (in time) existence result is proved in the energy space via Fourier analysis on compact Lie groups.
In this paper, a class of non-Newton filtration equations with singular potential and logarithmic nonlinearity under initial-boundary condition is investigated. Based on potential well method and Hardy-Sobolev inequality, the global existence of solutions is derived when the initial energy $J(u_0)$ is subcritical($J(u_0)<d$), critical($J(u_0)=d$) with $d$ being the mountain-pass level. Finite time blow-up results are obtained as well when the initial energy $J(u_0)$ satisfies specific conditions. Moreover, the upper and lower bounds of the blow-up time are given.
We consider in this paper blow-up solutions of the semilinear wave equation in one space dimension, with an exponential source term. Assuming that initial data are in $H^{1}_{loc}\times L^2_{loc}$ or some times in $ W^{1,\infty}\times L^{\infty}$, we derive the blow-up rate near a non-characteristic point in the smaller space, and give some bounds near other points. Our result generalize those proved by Godin under high regularity assumptions on initial data.
We consider the following nonlinear Schrödinger equation with a Hartree nonlinearity: \[ i\frac{\partial u}{\partial t}+Δu+|u|^{\frac{4}{N}}u\pm\left(\frac{1}{|x|^{2σ}}\star|u|^2\right)u=0 \] in $\mathbb{R}^N$. We are interested in the existence and behaviour of minimal mass blow-up solutions. Previous studies have shown the existence of minimal mass blow-up solutions with an inverse power potential and investigated the behaviour of the solution. In this paper, we investigate Hartree nonlinearity, which is a nonlinear term similar to the inverse power-type potential in terms of scaling.
We consider the semilinear heat equation $u_t=Δu+|u|^{p-1}u-|u|^{q-1}u$ in $\mathbb{R}^n\times(0,T)$, where $n=5$, $p=\frac{n+2}{n-2}$ and $q\in(0,1)$. By the presence of $-|u|^{q-1}u$, this equation has a finite time extinction property. We show the existence of a new type of blowup solutions by using this property. In fact, we obtain such blowup solutions by connecting a specific blowup solution of $u_t=Δu+|u|^{p-1}u$ and a specific solution of $u_t=Δu-|u|^{q-1}u$, and by adding correction terms.
In this note, we prove a blow-up result for a semilinear generalized Tricomi equation with nonlinear term of derivative type, i.e., for the equation $\mathscr{T}_{\!\!\ell} u = |\partial_t u|^p$, where $ \mathscr{T}_{\!\!\ell} = \partial_t^2-t^{2\ell}Δ$. Smooth solutions blow up in finite time for positive Cauchy data when the exponent $p$ of the nonlinear term is below $\frac{\mathscr{Q}}{\mathscr{Q}-2}$, where $\mathscr{Q}=(\ell+1)n+1$ is the quasi-homogeneous dimension of the generalized Tricomi operator $\mathscr{T}_{\!\!\ell}$. Furthermore, we get also an upper bound estimate for the life…
Blow-up and lifespan estimate for the generalized tricomi equation with the scale-invariant damping and time derivative nonlinearity on exterior domain
In this article, we investigate the blow-up for local solutions to a semilinear wave equation in the generalized Einstein - de Sitter spacetime with nonlinearity of derivative type. More precisely, we consider a semilinear damped wave equation with a time-dependent and not summable speed of propagation and with a time-dependent coefficient for the linear damping term with critical decay rate. We prove in this work that the results obtained in a previous work, where the damping coefficient takes two particular values $0$ or $2$, can be extended for any positive damping coefficient. In the blow-…
We consider the 1D nonlinear Schrödinger equation (NLS) with focusing \emph{point nonlinearity}, $$i\partial_tψ+ \partial_x^2ψ+ δ|ψ|^{p-1}ψ= 0$$ where $δ=δ(x)$ is the delta function supported at the origin. In the $L^2$ supercritical setting $p>3$, we construct self-similar blow-up solutions belonging to the energy space $L_x^\infty \cap \dot H_x^1$. This is reduced to finding outgoing solutions of a certain stationary profile equation. All outgoing solutions to the profile equation are obtained by using parabolic Weber functions and solving the jump condition at $x=0$ imposed by the $δ$ term.…
We consider in this article the weakly coupled system of wave equations in the \textit{scale-invariant case} and with time-derivative nonlinearities. Under the usual assumption of small initial data, we obtain an improvement of the delimitation of the blow-up region by obtaining a new candidate for the critical curve. More precisely, we enhance the results obtained in \cite{Palmieri} for the system under consideration in the present work. We believe that our result is optimal in the sense that beyond the blow-up region obtained here we may conjecture the global existence of the solution.
We study the impact of the convective terms on the global solvability or finite time blow up of solutions of dissipative PDEs. We consider the model examples of 1D Burger's type equations, convective Cahn-Hilliard equation, generalized Kuramoto-Sivashinsky equation and KdV type equations, we establish the following common scenario: adding sufficiently strong (in comparison with the destabilizing nonlinearity) convective terms to equation prevents the solutions from blowing up in finite time and makes the considered system globally well-posed and dissipative and for weak enough convective terms…
The final goal of this paper is to prove existence of local (strong) solutions to a (fully nonlinear) porous medium equation with blow-up term and nondecreasing constraint. To this end, the equation, arising in the context of Damage Mechanics, is reformulated as a mixed form of two different types of doubly nonlinear evolution equations. Global (in time) solutions to some approximate problems are constructed by performing a time discretization argument and by taking advantage of energy techniques based on specific structures of the equation. Moreover, a variational comparison principle for (po…
In this paper we consider the following biharmonic equation with critical exponent $P_ε$ : $Δ^2 u= Ku^{(n+4)/(n-4)-ε}, u>0$ in $Ω$ and $u=Δu=0$ on $\partialΩ$, where $Ω$ is a domain in $R^n$, $n\geq 5$, $ε$ is a small positive parameter and $K$ is smooth positive function. We construct solutions of $P_ε$ which blow up and concentrate at strict local maximum of $K$ either at the boundary or in the interior of $Ω$. We also construct solutions of $P_ε$ concentrating at an interior strict local minimum of $K$. Finally, we prove a nonexistense result for the corresponding supercritical problem whic…
We investigate the Cauchy problem for the focusing inhomogeneous nonlinear Schrödinger equation $i \partial_t u + Δu = - |x|^b |u|^{p-1} u$ in the radial Sobolev space $H^1_{\text{rad}}(\mathbb{R}^N)$, where $b>0$ and $p>1$. We show the global existence and energy scattering in the inter-critical regime, i.e., $p>\frac{N+4+2b}{N}$ and $p<\frac{N+2+2b}{N-2}$ if $N\geq 3$. We also obtain blowing-up solutions for the mass-critical and mass-supercritical nonlinearities. The main difficulty, coming from the spatial growing nonlinearity, is overcome by refined Gagliardo-Nirenberg type inequalities. …
Regularity and singularity of the blow-up curve for a wave equation with a derivative nonlinearity and a scale-invariant damping
University of Tunis El Manar（突尼斯曼纳尔大学）；Imam Abdulrahman Bin Faisal University（伊玛目阿卜杜勒拉赫曼·本·费萨尔大学）；Musashino University（武藏野大学）；Université Sorbonne Paris Nord（北索邦巴黎大学）
This paper is devoted to the lifespan of solutions to a damped fourth-order wave equation with logarithmic nonlinearity $$u_{tt}+Δ^2u-Δu-ωΔu_t+α(t)u_t=|u|^{p-2}u\ln|u|.$$ Finite time blow-up criteria for solutions at both lower and high initial energy levels are established, and an upper bound for the blow-up time is given for each case. Moreover, by constructing a new auxiliary functional and making full use of the strong damping term, a lower bound for the blow-up time is also derived.
In this note, we prove blow-up results for semilinear wave models with damping and mass in the scale-invariant case and with nonlinear terms of derivative type. We consider the single equation and the weakly coupled system. In the first case we get a blow-up result for exponents below a certain shift of the Glassey exponent. For the weakly coupled system we find as critical curve a shift of the corresponding curve for the weakly coupled system of semilinear wave equations with the same kind of nonlinearities. Our approach follows the one for the respective classical wave equation by Zhou Yi. I…
We consider a nonlinear Schrödinger equation with double power nonlinearity, where one power is focusing and mass critical and the other mass sub-critical. Classical variational arguments ensure that initial data with mass less than the mass of the ground state of the mass critical problem lead to global in time solutions. We are interested by the threshold dynamic and in particular by the existence of finite time blow up minimal solutions. For the mass critical problem, such an object exists thanks to the explicit conformal symmetry, and is in fact unique. For the focusing double power nonlin…
In this paper, we consider the initial boundary value problem for a pseudo-parabolic Kirchhoff equation with logarithmic nonlinearity. We use the potential well method to give a threshold result of global existence and finite-time blow-up for the weak solutions with initial energy $J(u_0)\leq d$. When the initial energy $J(u_0)>d$, we find another criterion for the vanishing solution and blow-up solution. We also get the exponential decay rate of the global solution and life span of the blow-up solution. Meanwhile, we study the corresponding stationary problem and establish a convergence relat…
In the present paper, we investigate the blow-up dynamics for local solutions to the semilinear generalized Tricomi equation with combined nonlinearity. As a result, we enlarge the blow-up region in comparison to the ones for the corresponding semilinear models with either power nonlinearity or nonlinearity of derivative type. Our approach is based on an iteration argument to establish lower bound estimates for the space average of local solutions. Finally, we obtain upper bound estimates for the lifespan of local solutions as byproduct of our iteration argument.
In this paper, a strongly damped semilinear wave equation with a general nonlinearity is considered. With the help of a newly constructed auxiliary functional and the concavity argument, a general finite time blow-up criterion is established for this problem. Furthermore, the lifespan of the weak solution is estimated from both above and below. This partially extends some results obtained in recent literatures and sheds some light on the similar effect of power type nonlinearity and logarithmic nonlinearity on finite time blow-up of solutions to such problems.
We consider the global regularity problem for nonlinear wave systems $$ \Box u = f(u) $$ on Minkowski spacetime ${\bf R}^{1+d}$ with d'Alambertian $\Box := -\partial_t^2 + \sum_{i=1}^d \partial_{x_i}^2$, where the field $u \colon {\bf R}^{1+d} \to {\bf R}^m$ is vector-valued, and the nonlinearity $f \colon {\bf R}^m \to {\bf R}^m$ is a smooth function with $f(0)=0$ and all derivatives bounded; the higher-dimensional sine-Gordon equation $\Box u = \sin u$ is a model example of this class of nonlinear wave system. For dimensions $d \leq 9$, it follows from the work of Heinz, Pecher, Brenner, and…
We consider a partial differential equation that arises in the coarse-grained description of epitaxial growth processes. This is a parabolic equation whose evolution is governed by the competition between the determinant of the Hessian matrix of the solution and the biharmonic operator. This model might present a gradient flow structure depending on the boundary conditions. We first extend previous results on the existence of stationary solutions to this model for Dirichlet boundary conditions. For the evolution problem we prove local existence of solutions for arbitrary data and global existe…
In this paper we investigate regularity properties of weak solutions to a PDE system that arises in the study of biological transport networks. The system consists of a possibly singular elliptic equation for the scalar pressure of the underlying biological network coupled to a diffusion equation for the conductance vector of the network. There are several different types of nonlinearities in the system. Of particular mathematical interest is a term that is a polynomial function of solutions and their partial derivatives and this polynomial function has degree three. That is, the system contai…
We consider the 1D nonlinear Schrödinger equation (NLS) with focusing point nonlinearity, $$ (δ\text{NLS}) \qquad i\partial_tψ+ \partial_x^2ψ+ δ|ψ|^{p-1}ψ= 0, $$ where $δ=δ(x)$ is the delta function supported at the origin. We show that $δ$NLS shares many properties in common with those previously established for the focusing autonomous translationally-invariant NLS $$ (\text{NLS}) \qquad i\partial_t ψ+ Δψ+ |ψ|^{p-1}ψ=0 \,. $$ The critical Sobolev space $\dot H^{σ_c}$ for $δ$NLS is $σ_c=\frac12-\frac{1}{p-1}$, whereas for NLS it is $σ_c=\frac{d}{2}-\frac{2}{p-1}$. In particular, the $L^2$ crit…
In this paper, we study the blow-up of solutions for semilinear wave equations with scale-invariant dissipation and mass in the case in which the model is somehow 'wave-like'. A Strauss type critical exponent is determined as the upper bound for the exponent in the nonlinearity in the main theorems. Two blow-up results are obtained for the sub-critical case and for the critical case, respectively. In both cases, an upper bound lifespan estimate is given.
It is shown that, for the bi-harmonic equation, an optimal regularity criterion of the vertex of typical paraboloids can be expressed in terms of Osgood-Dini integral conditions of Petrovskii's type for the heat equation derived in 1934. Some extensions of the first Fourier coefficent method to other PDEs are discussed. A survey on boundary point regularity for elliptic and parabolic PDEs is enclosed.
Finite time singularity formation in a fourth order nonlinear parabolic partial differential equation (PDE) is analyzed. The PDE is a variant of a ubiquitous model found in the field of Micro-Electro Mechanical Systems (MEMS) and is studied on a one-dimensional (1D) strip and the unit disc. The solution itself remains continuous at the point of singularity while its higher derivatives diverge, a phenomenon known as quenching. For certain parameter regimes it is shown numerically that the singularity will form at multiple isolated points in the 1D strip case and along a ring of points in the ra…
We consider a sequence of blowup solutions of a two dimensional, second order elliptic equation with exponential nonlinearity and singular data. This equation has a rich background in physics and geometry. In a work of Bartolucci-Chen-Lin-Tarantello it is proved that the profile of the solutions differs from global solutions of a Liouville type equation only by a uniformly bounded term. The present paper improves their result and establishes an expansion of the solutions near the blowup points with a sharp error estimate.
We consider the fractional Schrödinger equations with focusing Hartree type nonlinearity. When the energy is negative, we use a Glassey's virial type argument to show the finite time blow-up of solutions.
We present a notion of weak solution for the Dirichlet problem driven by the fractional Laplacian, following the Stampacchia theory. Then, we study semilinear problems on bounded domains $Ω$ with two different boundary conditions at the same time: the shape of the solution outside $Ω$ and a weighted limit to the boundary. We allow the nonlinearity to be positive or negative and we look for solutions blowing up at the boundary. Our starting observation is the existence of s-harmonic functions which explode at the boundary: these will be used both as supersolutions in the case of negative nonlin…
