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Let $\mathbb G$ be a graded Lie group with homogeneous dimension $Q$. In this paper, we study the Cauchy problem for a semilinear hypoelliptic damped wave equation involving a positive Rockland operator $\mathcal{R}$ of homogeneous degree $\nu\geq 2$ on $\mathbb G$ with power type nonlinearity $|u|^p$ and initial data taken from negative order homogeneous Sobolev space $\dot H^{-\gamma}(\mathbb G), \gamma>0,$ for the critical exponent case $p=1+\frac{2\nu}{Q+2\gamma}.$ We also explore the diffusion phenomenon of the higher-order hypoelliptic damped wave equations on graded Lie groups with initial data belonging to Sobolev spaces of negative order. We emphasize that our results are also new, even in the setting of higher-order differential operators on $\mathbb{R}^n$, and more generally, on stratified Lie groups., Comment: The previous incorrect version should be ignored, replaced by the current version, which is also done in the setting of general graded Lie groups. arXiv admin note: substantial text overlap with arXiv:2404.08766

We first study the existence, uniqueness and well-posedness of a general class of integro-differential diffusion equation on $L^p(\mathbb{G})$ $(1

We consider a higher order in (time) semilinear evolution inequality posed on the Kor\'{a}nyi ball under an inhomogeneous Dirichlet-type boundary condition. The problem involves an inverse-square potential $\lambda/|\xi|_\mathbb{H}^2$, where $\lambda \geq -(Q-2)^2/4$ and a general weight function $V$ depending on the space variable in front of the power nonlinearity. We first establish a general nonexistence result for the considered problem. Next, in the special case $V(\xi):=|\xi|_\mathbb{H}^a$, $a\in \mathbb{R}$, we prove the sharpness of our nonexistence result and show that the problem admits three different critical behaviors according to the value of the parameter $\lambda$., Comment: 24 pages

In the present paper, we consider the parabolic and hyperbolic inequalities with a singular potentials and with a critical nonlinearities in the annulus domain. The problems are studied with Neumann-type and Dirichlet-type boundary conditions on the boundary. Moreover, we study the systems of problems too. We have proved that the above problems are globally unsolvable in critical cases, thereby filling the gaps the recent results by Jleli and Samet in [J. Math. Anal. Appl. 514: 2 (2022)] and in [Anal. Math. Phys. 12: 90 (2022)]. Proofs are carried out using the method of test functions with logarithmic arguments, which is being developed for the first time in bounded domains.

We study a semilinear wave inequality with double damping on a complete noncompact Riemannian manifold. The considered problem involves a potential function $V$ depending on the space variable in front of the power nonlinearity and an inhomogeneous term $W$ depending on both time and space variables. Namely, we establish sufficient conditions for the nonexistence of weak solutions in both cases: $W\equiv 0$ and $W\not\equiv 0$. The obtained conditions depend on the parameters of the problem as well as the geometry of the manifold. Some special cases of manifolds, and of $V$ and $W$ are discussed in detail., Comment: 21 pages

The present paper considers the Cauchy-Dirichlet problem for the time-nonlocal reaction-diffusion equation $$\partial_t (k\ast(u-u_0))+\mathcal{L}_x [u]=f(u),\,\,\,\, x\in\Omega\subset\mathbb{R}^n, t>0,$$ where $k\in L^1_{loc}(\mathbb{R}_+),$ $f$ is a locally Lipschitz function, $\mathcal{L}_x$ is a linear operator. This model arises when studying the processes of anomalous and ultraslow diffusions. Results regarding the local and global existence, decay estimates, and blow-up of solutions are obtained. The obtained results provide partial answers to some open questions posed by Gal and Varma (2020), as well as Luchko and Yamamoto (2016). Furthermore, possible quasi-linear extensions of the obtained results are discussed, and some open questions are presented., Comment: 16 pages

The paper studies the large-time behavior of solutions to the Robin problem for PDEs with critical nonlinearities. For the considered problems, nonexistence results are obtained, which complements the interesting recent results by Ikeda et al. [J. Differential Equations, 269 (2020), no. 1, 563-594], where critical cases were left open. Moreover, our results provide partially answers to some other open questions previously posed by Zhang [Proc. Roy. Soc. Edinburgh Sect. A, 131 (2001), no. 2, 451-475] and Jleli-Samet [Nonlinear Anal., 178 (2019), 348-365]., Comment: 16 pages

In this paper, we consider the Cauchy problem for the degenerate parabolic equations on the Heisenberg groups with power law non-linearities. We obtain Fujita-type critical exponents, which depend on the homogeneous dimension of the Heisenberg groups. The analysis includes the case of porous medium equations. Our proof approach is based on methods of nonlinear capacity estimates specifically adapted to the nature of the Heisenberg groups. We also use the Kaplan eigenfunctions method in combination with the Hopf-type lemma on the Heisenberg groups., Comment: 34 pages

In this paper, we study the nonlinear Sobolev type equations on the Heisenberg group. We show that the problems do not admit nontrivial local weak solutions, i.e. "instantaneous blow up" occurs, using the nonlinear capacity method. Namely, by choosing suitable test functions, we will prove an instantaneous blow up for any initial conditions $u_0,\,u_1\in L^q(\mathbb{H}^n)$ with $q\leq \frac{Q}{Q-2}$., Comment: 14 pages

This paper studies the large-time behavior of solutions to the quasilinear inhomogeneous parabolic equation with combined nonlinearities. This equation is a natural extension of the heat equations with combined nonlinearities considered by Jleli-Samet-Souplet (Proc AMS, 2020). Firstly, we focus on an interesting phenomenon of discontinuity of the critical exponents. In particular, we will fill the gap in the results of Jleli-Samet-Souplet for the critical case. We are also interested in the influence of the forcing term on the critical behavior of the considered problem, so we will define another critical exponent depending on the forcing term., Comment: 12 pages

In this paper, the initial-boundary value problems for the time-fractional degenerate evolution equations are considered. Firstly, in the linear case, we obtain the optimal rates of decay estimates of the solutions. The decay estimates are also established for the time-fractional evolution equations with nonlinear operators such as: p-Laplacian, the porous medium operator, degenerate operator, mean curvature operator, and Kirchhoff operator. At the end, some applications of the obtained results are given to derive the decay estimates of global solutions for the time-fractional Fisher-KPP-type equation and the time-fractional porous medium equation with the nonlinear source., Comment: 23 pages. The previous version of the paper has been edited according to the comments of the reviewers

In this paper, we study a critical exponent to the semilinear heat equation with forcing term on Heisenberg group. Our technique of proof is based on methods of nonlinear capacity estimates specifically adapted to the nature of the Heisenberg group. Surprisingly, the critical exponent will be bounded for all dimensions of the Heisenberg group, in contrast to the Euclidean case which in the 1D and 2D cases will be infinite. In addition, we give an upper bound estimate on the lifespan of local solutions., Comment: 16 pages, we have improved the result

In the present paper, we study an inhomogeneous pseudo-parabolic equation with nonlocal nonlinearity $$u_t-k\Delta u_t-\Delta u=I^\gamma_{0+}(|u|^{p})+\omega(x),\,\ (t,x)\in(0,\infty)\times\mathbb{R}^N,$$ where $p>1,\,k\geq 0$, $\omega(x)\neq0$ and $I^\gamma_{0+}$ is the left Riemann-Liouville fractional integral of order $\gamma\in(0,1).$ Based on the test function method, we have proved the blow-up result for the critical case $\gamma=0,\,p=p_c$ for $N\geq3$, which answers an {\bf open question} posed in \cite{Zhou}, and in particular when $k=0$ it improves the result obtained in \cite{Bandle}. An interesting fact is that in the case $\gamma>0$, the problem does not admit global solutions for any $p>1$ and $\int_{\mathbb{R}^N}\omega(x) dx>0.$, Comment: 7 pages, comments are welcome

In the present paper, we study the Cauchy-Dirichlet problem to the nonlocal nonlinear diffusion equation with polynomial nonlinearities $$\mathcal{D}_{0|t}^{\alpha }u+(-\Delta)^s_pu=\gamma|u|^{m-1}u+\mu|u|^{q-2}u,\,\gamma,\mu\in\mathbb{R},\,m>0,q>1,$$ involving time-fractional Caputo derivative $\mathcal{D}_{0|t}^{\alpha}$ and space-fractional $p$-Laplacian operator $(-\Delta)^s_p$. We give a simple proof of the comparison principle for the considered problem using purely algebraic relations, for different sets of $\gamma,\mu,m$ and $q$. The Galerkin approximation method is used to prove the existence of a local weak solution. The blow-up phenomena, existence of global weak solutions and asymptotic behavior of global solutions are classified using the comparison principle., Comment: 33 pages

In this paper, we show the nonexistence results for the Kirchhoff elliptic, parabolic, and hyperbolic type equations on the Heisenberg groups. Also, the pseudo-parabolic and pseudo-hyperbolic equations of the Kirchhoff-type are under consideration. To prove these results we use the test function method. In addition, the analogous results are transferred to the cases of systems. Also, we give some examples of non-local nonlinearities., Comment: 37 pages

In this paper Tricomi-Gellerstedt-Keldysh-type fractional elliptic equations are studied. The results on the well-posedness of fractional elliptic boundary value problems are obtained for general positive operators with discrete spectrum and for Fourier multipliers with positive symbols. As examples, we discuss results in half-cylinder, star-shaped graph, half-space and other domains., Comment: 16 pages

In this paper in the cylindrical domain we consider a fractional elliptic operator with Dirichlet conditions. We prove, that the first eigenvalue of the fractional elliptic operator is minimised in a circular cylinder among all cylindrical domains of the same Lebesgue measure. This inequality is called the Rayleigh-Faber-Krahn inequality. Also, we give Lyapunov and Hartmann-Wintner inequalities for the fractional elliptic boundary value problem., Comment: 13 pages

The purpose of this paper is to establish the solvability results to direct and inverse problems for time-fractional pseudo-parabolic equations with the self-adjoint operators. We are especially interested in proving existence and uniqueness of the solutions in the abstract setting of Hilbert spaces., Comment: 18 pages

In this paper we obtain new estimates of the sequential Caputo fractional derivatives of a function at its extremum points. We derive comparison principles for the linear fractional differential equations, and apply these principles to obtain lower and upper bounds of solutions of linear and nonlinear fractional differential equations. The extremum principle is then applied to show that the initial-boundary-value problem for nonlinear anomalous diffusion possesses at most one classical solution and this solution depends continuously on the initial and boundary data. This answers positively to the open problem about maximum principle for the space and time-space fractional PDEs posed by Luchko in 2011. The extremum principle for an elliptic equation with a fractional derivative and for the fractional Laplace equation are also proved., Comment: 24 pages. arXiv admin note: substantial text overlap with arXiv:1806.04886

In the present paper initial problems for the semilinear integro-differential diffusion equation and system are considered. The analogue of Duhamel principle for the linear integro-differential diffusion equation is proved. The results on existence of local mild solutions and Fujita-type critical exponents to the semilinear integro-differential diffusion equation and system are presented., Comment: 27 pages

This paper is devoted to the study of initial-boundary value problems for time-fractional analogues of Korteweg-de Vries, Benjamin-Bona-Mahony, Burgers, Rosenau, Camassa-Holm, Degasperis-Procesi, Ostrovsky and time-fractional modified Korteweg-de Vries-Burgers equations on a bounded domain. Sufficient conditions for the blowing-up of solutions in finite time of aforementioned equations are presented. We also discuss the maximum principle and influence of gradient non-linearity on the global solvability of initial-boundary value problems for the time-fractional Burgers equation. The main tool of our study is the Pohozhaev nonlinear capacity method. We also provide some illustrative examples., Comment: 24 pages

A time-space fractional reaction-diffusion equation in a bounded domain is considered. Under some conditions on the initial data, we show that solutions may experience blow-up in a finite time. However, for realistic initial conditions, solutions are global in time. Moreover, the asymptotic behavior of bounded solutions is analysed., Comment: 6 pages, Quaestiones Mathematicae (2020)

A class of inverse problems for restoring the right-hand side of a parabolic equation for a large class of positive operators with discrete spectrum is considered. The results on existence and uniqueness of solutions of these problems as well as on the fractional time diffusion (subdiffusion) equations are presented. Consequently, the obtained results are applied for the similar inverse problems for a large class of subelliptic diffusion and subdiffusion equations (with continuous spectrum). Such problems are modelled by using general homogeneous left-invariant hypoelliptic operators on general graded Lie groups. A list of examples is discussed, including Sturm-Liouville problems, differential models with involution, fractional Sturm-Liouville operators, harmonic and anharmonic oscillators, Landau Hamiltonians, fractional Laplacians, and harmonic and anharmonic operators on the Heisenberg group. The rod cooling problem for the diffusion with involution is modelled numerically, showing how to find a "cooling function", and how the involution normally slows down the cooling speed of the rod., Comment: 26 pages, 7 figures. arXiv admin note: text overlap with arXiv:1812.01336

This paper deals with the multi-term generalisation of the time-fractional diffusion-wave equation for general operators with discrete spectrum, as well as for positive hypoelliptic operators, with homogeneous multi-point time-nonlocal conditions. Several examples of the settings where our nonlocal problems are applicable are given. The results for the discrete spectrum are also applied to treat the case of general homogeneous hypoelliptic left-invariant differential operators on general graded Lie groups, by using the representation theory of the group. For all these problems, we show the existence, uniqueness, and the explicit representation formulae for the solutions., Comment: 32 pages

In this paper we obtain new estimates of the Hadamard fractional derivatives of a function at its extreme points. The extremum principle is then applied to show that the initial-boundary-value problem for linear and nonlinear time-fractional diffusion equations possesses at most one classical solution and this solution depends continuously on the initial and boundary conditions. The extremum principle for an elliptic equation with a fractional Hadamard derivative is also proved.

In this paper, a maximum principle for the one-dimensional sub-diffusion equation with Atangana-Baleanu fractional derivative is formulated and proved. The proof of the maximum principle is based on an extremum principle for the Atangana-Baleanu fractional derivative that is given in the paper, too. The maximum principle is then applied to show that the initial-boundary-value problem for the linear and nonlinear time-fractional diffusion equations possesses at most one classical solution and this solution continuously depends on the initial and boundary conditions., Comment: to appear in Applied Mathematics Letters

A class of inverse problems for a heat equation with involution perturbation is considered using four different boundary conditions, namely, Dirichlet, Neumann, periodic and anti-periodic boundary conditions. Proved theorems on existence and uniqueness of solutions to these problems are presented. Solutions are obtained in the form of series expansion using a set of appropriate orthogonal basis for each problem. Convergence of the obtained solutions is also discussed., Comment: 17 pages

In this paper we consider a non-local problem for a Laplace operator in a multidimensional bounded symmetric domain. The investigated problem is an analogue of the classical periodic boundary value problems in the case of non-rectangular domain. We prove self-adjointness of the problem and show a method of constructing eigenfunctions. We obtain an analogue of the Rayleigh type inequality and some spectral inequalities for the first eigenvalue of the nonlocal problem., Comment: 7 pages

In this paper we consider the Cauchy problem for multidimensional elliptic equations in a cylindrical domain. The method of spectral expansion in eigenfunctions of the Cauchy problem for equations with deviating argument establishes a criterion of the strong solvability of the considered elliptic Cauchy problem. It is shown that the ill-posedness of the elliptic Cauchy problem is equivalent to the existence of an isolated point of the continuous spectrum for a self-adjoint operator with deviating argument., Comment: 10 pages

In this paper we study some boundary value problems for a fractional analogue of second order elliptic equation with an involution perturbation in a rectangular domain. Theorems on existence and uniqueness of a solution of the considered problems are proved by spectral method., Comment: 10 pages