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In this paper, we proposed a new method to prove the blowup at $ +\infty $ for the solution of the initial boundary value problem for a class of semilinear parabolic equations with logarithmic nonlinearity at sub-critical initial energy level by contradiction. Moreover, following the idea of contradiction, we obtained a sufficient criterion for the blowup at $ +\infty $ of the solution with arbitrarily positive initial energy by defining a new auxiliary function, in which the location of the initial data concerning the unstable set is not required. [ABSTRACT FROM AUTHOR]

In the conventional two-grid (TG) method, the nonlinear system on the fine grid is transformed into a nonlinear subsystem on the coarse grid and a linear subsystem on the fine grid to reduce computational costs. It has been successfully applied in various fields. Nonetheless, its computational efficiency remains relatively low. For this, we develop a novel reduced-order two-grid (ROTG) method with less degrees of freedom for solving the semilinear parabolic equation. For the two subsystems mentioned, the proper orthogonal decomposition (POD) technique is utilized to substantially reduce degrees of freedom. An a priori error estimate for the ROTG scheme is derived. Finally, we conduct several numerical tests to observe the ROTG method's behavior and verify the theoretical analysis. [ABSTRACT FROM AUTHOR]

Second order semilinear parabolic equation ∂u/ ∂t = div(A(x, t)∇u) + h(x, t, u) with time-periodic coefficients is considered in domain Ω × (−∞, +∞), where Ω is the exterior of a compact set in Rn x. Depending on the behavior of the function h(x, t, u) at infinity, the conditions are found under which the positive periodic solution does not exist. [ABSTRACT FROM AUTHOR]

In this paper, exact solutions of semilinear equations having exponential growth in the space variable x are found. Semilinear Schrödinger equation with logarithmic nonlinearity and third-order evolution equations arising in optics with logarithmic and power-logarithmic nonlinearities are investigated. In the parabolic case, the solution u is written as u = b e − a x 2 , a < 0 , a , b being real-valued functions. We are looking for the solutions u of Schrödinger-type equation of the form u = b e − a x 2 2 , respectively, for the third-order PDE, u = A e i Φ , where the amplitude b and the phase function a are complex-valued functions, A > 0 , and Φ is real-valued. In our proofs, the method of the first integral is used, not Hirota's approach or the method of simplest equation. [ABSTRACT FROM AUTHOR]

In this paper, we present a unified method for deriving differential Harnack inequalities for positive solutions of the semilinear parabolic equation ∂ t u = Δ V u + H (u) on complete Riemannian manifolds with Bakry-Émery curvature bounded below. This method transforms the problem of deriving differential Harnack inequalities into solving a related ODE system. As its application, on the one hand, we obtain new and improved estimates for logarithmic type equation and Yamabe type equation. On the other hand, we establish some sharp estimates for them under non-negative Bakry-Émery curvature condition. As their natural corollary, we also obtain some sharp Harnack inequalities and Liouville type theorems. [ABSTRACT FROM AUTHOR]

The time fractional order differential equations are fundamental tools that are used for modeling neuronal dynamics. These equations are obtained by substituting the time derivative of order α, where 0 < α < 1, in the standard equation with the Caputo fractional formula. In this paper, two implicit difference schemes: the linearly Euler implicit and the Crank-Nicolson (CN) finite difference schemes, are employed in solving a one-dimensional time-fractional semilinear equation with Dirichlet boundary conditions. Moreover, the consistency, stability and convergence of the proposed schemes are investigated. We prove that the IEM is unconditionally stable, while CNM is conditionally stable. Furthermore, a comparative study between these two schemes will be conducted via numerical experiments. The efficiency of the proposed schemes in terms of absolute errors, order of accuracy and computing time will be reported and discussed. [ABSTRACT FROM AUTHOR]

In this article, we discuss one of the subsections of finite element method (FEM), classified as the Composite Finite Element Method, abbreviated as CFE. Dimensionality reduction is the primary benefit of the CFE method as it helps to reduce the complexity for the domain space. The degrees of freedom is more in FEM, while compared to the CFE method. Here, the semilinear parabolic problem in a 2D convex polygonal domain is considered. The analysis of the semidiscrete method for the problem is carried out initially in the CFE framework. Here, the discretization would be carried out for the space co-ordinate. Then, fully discrete problem is taken into account, where both the spatial and time components get discretized. In the fully discrete case, the backward Euler method and the Crank-Nicolson method in the CFE framework is adapted for the semilinear problem. The properties of convergence are derived and the error estimates are examined. It is verified that the order of convergence is preserved. The results obtained from the numerical computations are also provided.

In this paper, exact solutions of semilinear equations having exponential growth in the space variable x are found. Semilinear Schrödinger equation with logarithmic nonlinearity and third-order evolution equations arising in optics with logarithmic and power-logarithmic nonlinearities are investigated. In the parabolic case, the solution u is written as u=be−ax2, a<0, a,b being real-valued functions. We are looking for the solutions u of Schrödinger-type equation of the form u=be−ax22, respectively, for the third-order PDE, u=AeiΦ, where the amplitude b and the phase function a are complex-valued functions, A>0, and Φ is real-valued. In our proofs, the method of the first integral is used, not Hirota’s approach or the method of simplest equation.

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This paper investigates the exact controllability of a semilinear reaction–diffusion equation governed by a multiplicative control. Here, the reaction term leads to an unbounded control operator. We will first study the well-posedness and then proceed to the question of exact steering for the system at hand. Our approaches are based on a fixed point argument and the linearization method. The obtained results are further analysed through a numerical example. [ABSTRACT FROM AUTHOR]

In this work, we study the existence, uniqueness and regularity of the solution to a semilinear Cauchy–Dirichlet problem with variable coefficients, using the Faedo–Galerkin method. We applied this study to a nonhomogeneous Burgers problem considered in a nonparabolic domain. We give a new regularity result of the solution in an anisotropic Sobolev space. [ABSTRACT FROM AUTHOR]

We formulate a quasilinear parabolic equation describing the behavior of the global-in-time solution to a semilinear parabolic equation. We study this equation in accordance with the blow-up and quenching patterns of the solution to the original semilinear parabolic equation. This quasilinear equation is new in the theory of partial differential equations and presents several difficulties for mathematical analysis. Two approaches are examined: functional analysis and a viscosity solution. [ABSTRACT FROM AUTHOR]

In this paper, we first investigate optimal control problem for semilinear parabolic and introduce the standard $ L^2(\Omega) $-orthogonal projection and the elliptic projection. Then we present some necessary intermediate variables and their error estimates. At last, we derive the error estimates between the finite element solutions and $ L^2 $-orthogonal projection or the elliptic projection of the exact solutions.

In this paper, we use the self-similar transformation and the modified potential well method to study the long time behaviors of solutions to the classical semilinear parabolic equation associated with critical Sobolev exponent in $\mathbb{R}^N$. Global existence and finite time blowup of solutions are proved when the initial energy is in three cases. When the initial energy is low or critical, we not only give a threshold result for the global existence and blowup of solutions, but also obtain the decay rate of the $L^2$ norm for global solutions. When the initial energy is high, sufficient conditions for the global existence and blowup of solutions are also provided. We extend the recent results which were obtained in [R. Ikehata, M. Ishiwata, T. Suzuki, Ann. Inst. H. Poincaré Anal. Non Linéaire 27(2010), No. 3, 877–900].

We consider an inverse problem for the determination of a purely time-dependent source in a semilinear parabolic equation with a nonlocal boundary condition. An approximation scheme for the solution together with the well-posedness of the problem with the initial value u0 ∈ H1(Ω) is presented by means of the Rothe time-discretization method. Further approximation scheme via Rothe's method is constructed for the problem when u0 ∈ L2(Ω) and the integral kernel in the nonlocal boundary condition is symmetric. [ABSTRACT FROM AUTHOR]

We formulate a quasilinear parabolic equation describing the behavior of the global-in-time solution to a semilinear parabolic equation. We study this equation in accordance with the blow-up and quenching patterns of the solution to the original semilinear parabolic equation. This quasilinear equation is new in the theory of partial differential equations and presents several difficulties for mathematical analysis. Two approaches are examined: functional analysis and a viscosity solution.

The main aim of this paper is to obtain superconvergence result for a semilinear parabolic equation with 3-step backward differential formula Galerkin finite element method. The time-discrete system is established to split the error into the temporal error and spatial error. The initial two steps of the temporal error is dealt with by a new way to ensure the third-order accuracy of the scheme. Some new tricks are utilized to get the spatial error in H 1 -norm of order O (h (h + τ 3 2 )) without the ratio between the spatial subdivision parameter h and the temporal step τ, which improves the corresponding results in the previous literature. The final superconvergence result is deduced by the above achievements and trigonometric inequality. Two numerical examples are provided to support the theoretical analysis. [ABSTRACT FROM AUTHOR]

We consider multiobjective optimal control problems for semilinear parabolic systems subject to pointwise state constraints, integral state-control constraints and pointwise state-control constraints. In addition, the data of the problems need not be twice Fréchet differentiable. Employing the second-order directional derivative (in the sense of Demyanov-Pevnyi) for the involved functions, we establish necessary optimality conditions, via second-order Lagrange multiplier rules of Fritz-John type, for local weak Pareto solutions of the problems. [ABSTRACT FROM AUTHOR]

Based on the research on the predator–prey model with Holling type response function, a delayed predator–prey system with diffusion term and habitat complexity effect is established, and the effects of time delay and diffusion on dynamical behavior of the system are studied. First, taking habitat complexity as the parameter, the dynamical properties of the system without time delay are studied. By eigenvalue analysis, the sufficient conditions for locally asymptotic stability of the positive equilibrium and globally asymptotic stability of the boundary equilibrium are given, the existence conditions of Hopf bifurcation induced by diffusion term are discussed. In an appropriate range, diffusion makes a family of spatially homogeneous and inhomogeneous periodic solutions bifurcate from the positive equilibrium. Second, taking production delay as the bifurcation parameter, the existence conditions of Hopf bifurcation are given, the method to determine the bifurcation direction and the stability of bifurcating periodic solutions is given by using the center manifold theory and normal form method. Finally, the biological interpretations of the results are given, and some numerical simulations are given to verify the theoretical analysis results. [ABSTRACT FROM AUTHOR]

We consider semilinear parabolic equations with nonlinear boundary conditions. We give conditions which guarantee global existence of solutions as well as blow-up in finite time of all solutions with nontrivial initial data. The results depend on the behavior of variable coefficients as $t \to \infty$.

In this paper, we use the self-similar transformation and the modified potential well method to study the long time behaviors of solutions to the classical semilinear parabolic equation associated with critical Sobolev exponent in RN. Global existence and finite time blowup of solutions are proved when the initial energy is in three cases. When the initial energy is low or critical, we not only give a threshold result for the global existence and blowup of solutions, but also obtain the decay rate of the L² norm for global solutions. When the initial energy is high, sufficient conditions for the global existence and blowup of solutions are also provided. We extend the recent results which were obtained in [R. Ikehata, M. Ishiwata, T. Suzuki, Ann. Inst. H. Poincaré Anal. Non Linéaire 27(2010), No. 3, 877-900]. [ABSTRACT FROM AUTHOR]

The use of external enzymes provides an alternative way of reducing communication in pathogenic bacteria that may lead to the degradation of their signal and the loss of their pathogeneity. The present study considers an optimal control problem for the semilinear reaction-diffusion model of bacterial quorum sensing under the impact of external enzymes. Estimates of the solution of the controlled system are obtained, on the basis of which the solvability of the extremal problem is proved and the necessary optimality conditions of the first-order are derived. A numerical algorithm to find a solution of the optimal control problem is constructed and implemented. The conducted numerical experiments demonstrate an opportunity to build an effective strategy of the enzymes impact for treatment. [ABSTRACT FROM AUTHOR]

In this paper, the initial boundary value problem for a nonlocal semilinear pseudo-parabolic equation is investigated, which was introduced to model phenomena in population dynamics and biological sciences where the total mass of a chemical or an organism is conserved. The existence, uniqueness and asymptotic behavior of the global solution and the blowup phenomena of solution with subcritical initial energy are established. Then these results are extended parallelly to the critical initial energy. Further the blowup phenomena of solution with supercritical initial energy is proved, but the existence, uniqueness and asymptotic behavior of the global solution with supercritical initial energy are still open.

We study a semilinear parabolic equation that possesses global bounded weak solutions whose gradient has a singularity in the interior of the domain for all t > 0. The singularity of these solutions is of the same type as the singularity of a stationary solution to which they converge as t → ∞.

Dynamical reconstruction of unknown time-varying controls from inexact measurements of the state function is investigated for a semilinear parabolic equation with memory. This system includes as particular cases the Schlögl model and the FitzHugh–Nagumo equations. A numerical method is suggested that is based on techniques of feedback control. An error analysis is performed. Numerical examples confirm the theoretical predictions.

Abstract In this paper, we investigate the condition ( C p ) α ∫ 0 u f ( s ) d s ≤ u f ( u ) + β u p + γ , u > 0 $$(C_{p})\quad \alpha \int _{0}^{u}f(s)\,ds \leq uf(u)+\beta u^{p}+\gamma ,\quad u>0 $$ for some α > 2 $\alpha >2$ , γ > 0 $\gamma >0$ , and 0 ≤ β ≤ ( α − p ) λ p , 0 p $0\leq \beta \leq \frac{ (\alpha -p ) \lambda _{p,0}}{p}$ , where p > 1 $p>1$ , and λ p , 0 $\lambda _{p,0}$ is the first eigenvalue of the discrete p-Laplacian Δ p , ω $\Delta _{p,\omega }$ . Using this condition, we obtain blow-up solutions to discrete p-Laplacian parabolic equations { u t ( x , t ) = Δ p , ω u ( x , t ) + f ( u ( x , t ) ) , ( x , t ) ∈ S × ( 0 , + ∞ ) , μ ( z ) ∂ u ∂ p n ( x , t ) + σ ( z ) | u ( x , t ) | p − 2 u ( x , t ) = 0 , ( x , t ) ∈ ∂ S × [ 0 , + ∞ ) , u ( x , 0 ) = u 0 ≥ 0 ( nontrivial ) , x ∈ S , $$ \textstyle\begin{cases} u_{t} (x,t )=\Delta _{p,\omega }u (x,t )+f(u(x,t)), & (x,t )\in S\times (0,+\infty ), \\ \mu (z)\frac{\partial u}{\partial _{p} n}(x,t)+\sigma (z) \vert u(x,t) \vert ^{p-2}u(x,t)=0, & (x,t )\in \partial S\times [0,+\infty ), \\ u (x,0 )=u_{0}\geq 0\quad (\mbox{nontrivial}), & x\in S, \end{cases} $$ on a discrete network S, where ∂ u ∂ p n $\frac{\partial u}{\partial _{p}n}$ denotes the discrete p-normal derivative. Here μ and σ are nonnegative functions on the boundary ∂S of S with μ ( z ) + σ ( z ) > 0 $\mu (z)+\sigma (z)>0$ , z ∈ ∂ S $z\in \partial S$ . In fact, we will see that condition ( C p ) $(C_{p})$ improves the conditions known so far.

We study the existence, uniqueness and the asymptotic behavior of solutions of a class of retarded non-autonomous semilinear parabolic equations with infinite delay effects in the weighted space C, nonlinearity containing super-linear perturbations, and time-dependent external forces. With a suitable exponential growth of the external force, we prove the existence of pullback attractors without restriction on the growth order of polynomial type nonlinearity. [ABSTRACT FROM AUTHOR]

The ubiquity of semilinear parabolic equations is clear from their numerous applications ranging from physics and biology to materials and social sciences. In this paper, we consider a practically desirable property for a class of semilinear parabolic equations of the abstract form ut = u + f[u], with a linear dissipative operator and f a nonlinear operator in space, namely, a time-invariant maximum bound principle, in the sense that the timedependent solution u preserves for all time a uniform pointwise bound in absolute value imposed by its initial and boundary conditions. We first study an analytical framework for sufficient conditions on and f that lead to such a maximum bound principle for the time-continuous dynamic system of infinite or finite dimensions. Then we utilize a suitable exponential time-differencing approach with a properly chosen generator of the contraction semigroup to develop first- and second-order accurate temporal discretization schemes that satisfy the maximum bound principle unconditionally in the time-discrete setting. Error estimates of the proposed schemes are derived along with their energy stability. Extensions to vector- and matrix-valued systems are also discussed. We demonstrate that the abstract framework and analysis techniques developed here offer an effective and unified approach to studying the maximum bound principle of the abstract evolution equation that covers a wide variety of well-known models and their numerical discretization schemes. Some numerical experiments are also carried out to verify the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, a 2‐step backward differentiation formula (BDF2) Galerkin method is investigated for semilinear parabolic equation. More precisely, the second‐order time‐stepping scheme is used for time discretization and the piecewise linear continuous Galerkin method is employed for spatial discretization, respectively. Optimal error estimates in L2 and H1‐norms are obtained without any restriction on the time‐step size, while previous works always require certain conditions on time step‐size. The key to our analysis is to derive a uniform boundness of the numerical solution in energy norm so as to avoid the inverse inequality used in the usual convergence analysis of the finite element methods. Numerical experiments are carried out to confirm the theoretical analysis. [ABSTRACT FROM AUTHOR]

The localized exponential time differencing method based on overlapping domain decomposition has been recently introduced and successfully applied to parallel computations for extreme-scale numerical simulations of coarsening dynamics based on phase field models. In this paper, we focus on numerical solutions of a class of semilinear parabolic equations with the well-known Allen–Cahn equation as a special case. We first study the semi-discrete system under the standard central difference spatial discretization and prove the equivalence between the monodomain problem and the corresponding multidomain problem obtained by the Schwarz waveform relaxation iteration. Then we develop the fully discrete localized exponential time differencing schemes and, by establishing the maximum bound principle, prove the convergence of the fully discrete localized solutions to the exact semi-discrete solution and the convergence of the iterative solutions. Numerical experiments are carried out to verify the theoretical results in one-dimensional space and test the convergence and accuracy of the proposed algorithms with different numbers of subdomains in two-dimensional space. [ABSTRACT FROM AUTHOR]

We study the existence and nonexistence of global weak solutions to the semilinear parabolic differential inequality ∂ t u - Δ u ≥ | u | p , (t , x) ∈ (0 , ∞) × B c , where p > 1 , B is the closed unit ball in R N ( N ≥ 2 ) and B c is its complement, under the semilinear dynamical boundary conditions ∂ t u + u ≥ | u | q + w (x) , (t , x) ∈ (0 , ∞) × ∂ B or ∂ t u + ∂ ν u + α u ≥ | u | q + w (x) , (t , x) ∈ (0 , ∞) × ∂ B , where q > 1 , α ≥ 0 , ∂ ν : = ∂ ∂ ν + , ν + is the outward unit normal (relative to B c ) on ∂ B and w ∈ L 1 (∂ B) , ∫ ∂ B w (x) d S x ≥ 0 . The cases ∫ ∂ B w (x) d S x = 0 and ∫ ∂ B w (x) d S x > 0 are discussed separately. [ABSTRACT FROM AUTHOR]

In this paper, the initial boundary value problem for a nonlocal semilinear pseudo-parabolic equation is investigated, which was introduced to model phenomena in population dynamics and biological sciences where the total mass of a chemical or an organism is conserved. The existence, uniqueness and asymptotic behavior of the global solution and the blowup phenomena of solution with subcritical initial energy are established. Then these results are extended parallelly to the critical initial energy. Further the blowup phenomena of solution with supercritical initial energy is proved, but the existence, uniqueness and asymptotic behavior of the global solution with supercritical initial energy are still open. [ABSTRACT FROM AUTHOR]

This paper addresses local stabilization of a semilinear parabolic system by boundary control. Under a certain assumption on the nonlinearity of the parabolic equation, a linear boundary feedback control law is applied to control the semilinear system. Based on the operator theories and relations of different norms, locally exponential stabilization of the closed loop system is established. Simulations support the established stability result. [ABSTRACT FROM AUTHOR]

We consider a semilinear parabolic equation with flux at the boundary governed by a nonlinear memory. We give some conditions for this problem which guarantee global existence of solutions as well as blow-up in finite time of all nontrivial solutions. The results depend on the behavior of variable coefficients as t → ∞. [ABSTRACT FROM AUTHOR]

We investigate the asymptotic behavior of solutions to a semilinear heat equation with homogeneous Neumann boundary conditions. It was recently shown that the nontrivial kernel of the linear part leads to the coexistence of fast solutions decaying to 0 exponentially (as time goes to infinity), and slow solutions decaying to 0 as negative powers of t. Here we provide a characterization of slow/fast solutions in terms of their sign, and we show that the set of initial data giving rise to fast solutions is a graph of codimension one in the phase space.

This article studies the existence and nonexistence of global solutions to the initial boundary value problems for semilinear wave and heat equation, and for Cauchy problem of nonlinear Schrodinger equation. This is done under three possible initial energy levels, except the NLS as it does not have comparison principle. The most important feature in this article is a new hypothesis on the nonlinear source terms which can include at least eight important and popular power-type nonlinearities as special cases. This article also finds some kinds of divisions for the initial data to guarantee the global existence or finite time blowup of the solution of the above three problems.

We study the behavior of solutions of the semilinear problem $$\displaylines{ u_t=u_{xx}+(1+(T-t)^{-\alpha}\chi_{\{|x|0$ and p>0 We describe, in terms of the parameters when the solution is bounded and when it blows up. For blowing up solutions we find the blow-up rate and the blow-up set.

We consider semilinear parabolic equations with nonlinear boundary conditions. We give conditions which guarantee global existence of solutions as well as blow-up in finite time of all solutions with nontrivial initial data. The results depend on the behavior of variable coefficients as t → ∊. [ABSTRACT FROM AUTHOR]

This study was conducted to present error analysis of a finite element method for computing the radially symmetric solutions of semilinear heat equations. Particularly, this study establishes optimal order error estimates in L ∞ and weighted L 2 norms, respectively, for the symmetric and nonsymmetric formulation. Some numerical examples are presented to validate the obtained theoretical results. [ABSTRACT FROM AUTHOR]

We construct finite time blow-up solutions to the 3-dimensional harmonic map flow into the sphere S2, ut = Δu + |∇u|2u in Ω×(0,T) u = ub on ∂Ω × (0,T )u(⋅,0) = u0 in Ω, with u(x,t): Ω × [0,T) → S2. Here Ω is a bounded, smooth axially symmetric domain in R3. We prove that for any circle Γ ⊂ Ω with the same axial symmetry, and any sufficiently small T > 0 there exist initial and boundary conditions such that u(x,t) blows-up exactly at time T and precisely on the curve Γ, in fact |∇u(⋅,t)|2 ⇀ |∇u∗|2 + 8πδΓ as t→T. for a regular function u∗(x), where δΓ denotes the Dirac measure supported on the curve. This the first example of a blow-up solution with a space-codimension 2 singular set, the maximal dimension predicted in the partial regularity theory by Chen-Struwe and Cheng [5,6]. [ABSTRACT FROM AUTHOR]

In this paper, we analyze optimal control problems of semilinear parabolic equations, where the controls are distributed and depend only on time. Box constraints for the controls are imposed and the cost functional does not involve the control itself, only the associated state. We prove second order optimality conditions for local strong minimizers, which are used to derive error estimates in the numerical approximation. First we estimate the difference between the discrete and continuous optimal states. In the last part, under an additional assumption on the optimal adjoint state, we prove error estimates for the controls and improve the estimates for the states. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we consider the following discrete p -Laplacian nonlinear parabolic equations with q -nonlocal reaction { u t (x , t) = Δ p , ω u (x , t) + λ ∑ y ∈ S | u (y , t) | q − 1 u (y , t) , (x , t) ∈ S × (0 , ∞) , u (x , t) = 0 , (x , t) ∈ ∂ S × (0 , ∞) , u (x , 0) = u 0 ≥ 0 , x ∈ S ‾. Here, S is a network with boundary ∂ S. The goal of this paper is to characterize completely the parameters p > 1 and q > 0 to see when the solution blows up, vanishes, or exists globally. Indeed, the blow-up rates for the blow-up solutions are derived. [ABSTRACT FROM AUTHOR]

A class of semilinear parabolic reaction diffusion equations with multiple time delays is considered. These time delays and corresponding weights are to be optimized such that the associated solution of the delay equation is the best approximation of a desired state function. The differentiability of the mapping is proved that associates the solution of the delay equation to the vector of weights and delays. Based on an adjoint calculus, first-order necessary optimality conditions are derived. Numerical test examples show the applicability of the concept of optimizing time delays. [ABSTRACT FROM AUTHOR]

A problem of sparse optimal boundary control for a semilinear parabolic partial differential equation is considered, where pointwise bounds on the control and mixed pointwise control-state constraints are given. A standard quadratic objective functional is to be minimized that includes a Tikhonov regularization term and the L1-norm of the control accounting for the sparsity. Applying a recent linearization theorem, we derive first-order necessary optimality conditions in terms of a variational inequality under linearized mixed control state constraints. Based on this preliminary result, a Lagrange multiplier rule with bounded and measurable multipliers is derived and sparsity results on the optimal control are demonstrated. [ABSTRACT FROM AUTHOR]

We study stability of stationary solutions for a class of nonlocal semilinear parabolic equations. To this end, we prove the Feynman–Kac formula for a Lévy processes with time-dependent potentials and arbitrary initial condition. We propose sufficient conditions for asymptotic stability of the zero solution, and use them to the study of the spatial logistic equation arising in population ecology. For this equation, we find conditions which imply that its positive stationary solution is asymptotically stable. We consider also the case when the initial condition is given by a random field. [ABSTRACT FROM AUTHOR]

We establish a new finite time blowup theorem for the solution to the initial boundary value problem of a class of semilinear pseudo-parabolic equations at high initial energy level. And we also estimate the upper bound of the blowup time. [ABSTRACT FROM AUTHOR]

Second order semilinear parabolic equation with time-periodic coefficients is considered in the domain {x; ∣x∣ > R}×(-∞++∞).The absence of global positive periodic solutions is studied. The exact conditions are found under which the positive periodic solution does not exist. [ABSTRACT FROM AUTHOR]

We give a new stability estimate for the problem of determining the time-dependent zero order coefficient in a parabolic equation from a partial parabolic Dirichlet-to-Neumann map. The novelty of our result is that, contrary to the previous works, we do not need any measurement on the final time. We also show how this result can be used to establish a stability estimate for the problem of determining the nonlinear term in a semilinear parabolic equation from the corresponding “linearized” Dirichlet-to-Neumann map. This is the first result for this kind of inverse problems. The key ingredient in our analysis is a parabolic version of an elliptic Carleman inequality due to Bukhgeim and Uhlmann [11] . This parabolic Carleman inequality enters in an essential way in the construction of complex geometric optic (abbreviated to CGO in this text) solutions that vanish at a part of the lateral boundary and initial or final time. [ABSTRACT FROM AUTHOR]

In the present paper, we study the initial boundary value problem of the sublinear parabolic equation. We prove the existence of solutions and investigate the stability and instability of stationary solutions. We show that a unique positive and a unique negative stationary solutions are exponentially stable and give the exact exponent. We prove that small stationary solutions are unstable. For one space dimensional autonomous equations, we elucidate the structure of stationary solutions and study the stability of all stationary solutions. [ABSTRACT FROM AUTHOR]

We study the behavior of solutions of the semilinear problem ut = uxx + (1 + (T - t) -αχ{|x|<(T -t) 1/2} )u p , x ϵ R, t ϵ (0, T), u(x, 0) = u0(x) ≥ 0, x ϵ R, with α > 0 and p > 0. We describe, in terms of the parameters when the solution is bounded and when it blows up. For blowing up solutions we find the blow-up rate and the blow-up set. [ABSTRACT FROM AUTHOR]