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Journal articles on the topic 'Semilinear heat equation'

Author: Grafiati

Published: 4 June 2021

Last updated: 26 July 2025

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1

Chill, Ralph. "Convergence of bounded solutions to gradient‐like semilinear Cauchy problems with radial nonlinearity." Asymptotic Analysis 33, no. 2 (2003): 93–106. https://doi.org/10.3233/asy-2003-524.

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We prove convergence to a steady state of bounded solutions of the abstract first order semilinear Cauchy problem ut+Lu+g(Ψ(u))Cu=0, t∈R+, and of the second order semilinear Cauchy problem utt+αut+Lu+g(Ψ(u))Cu=0, t∈R+. We apply the abstract results to semilinear parabolic and hyperbolic partial differential equations including the heat equation, the wave equation, a Kuramoto–Sivashinsky model and the Kirchhoff–Carrier equation.

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2

CAZENAVE, THIERRY, FLÁVIO DICKSTEIN, and FRED B. WEISSLER. "NON-REGULARITY IN HÖLDER AND SOBOLEV SPACES OF SOLUTIONS TO THE SEMILINEAR HEAT AND SCHRÖDINGER EQUATIONS." Nagoya Mathematical Journal 226 (September 9, 2016): 44–70. http://dx.doi.org/10.1017/nmj.2016.35.

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In this paper, we study the Cauchy problem for the semilinear heat and Schrödinger equations, with the nonlinear term $f(u)=\unicode[STIX]{x1D706}|u|^{\unicode[STIX]{x1D6FC}}u$. We show that low regularity of $f$ (i.e., $\unicode[STIX]{x1D6FC}>0$ but small) limits the regularity of any possible solution for a certain class of smooth initial data. We employ two different methods, which yield two different types of results. On the one hand, we consider the semilinear equation as a perturbation of the ODE $w_{t}=f(w)$. This yields, in particular, an optimal regularity result for the semilinear

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3

Barbu, V., and G. Wang. "Feedback stabilization of semilinear heat equations." Abstract and Applied Analysis 2003, no. 12 (2003): 697–714. http://dx.doi.org/10.1155/s1085337503212100.

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This paper is concerned with the internal and boundary stabilization of the steady-state solutions to quasilinear heat equations via internal linear feedback controllers provided by an LQ control problem associated with the linearized equation.

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4

Saanouni, Tarek. "Global Well-Posedness and Finite-Time Blow-Up of Some Heat-Type Equations." Proceedings of the Edinburgh Mathematical Society 60, no. 2 (2016): 481–97. http://dx.doi.org/10.1017/s0013091516000213.

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AbstractWe study two different heat-type equations. First, global well-posedness in the energy space of some high-order semilinear heat-type equation with exponential nonlinearity is obtained for even space dimensions. Second, a finite-time blow-up result for the critical monomial focusing heat equation with the p-Laplacian is proved.

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5

Wang, Lijuan, and Can Zhang. "A uniform bound on costs of controlling semilinear heat equations on a sequence of increasing domains and its application." ESAIM: Control, Optimisation and Calculus of Variations 28 (2022): 8. http://dx.doi.org/10.1051/cocv/2022001.

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In this paper, we first prove a uniform upper bound on costs of null controls for semilinear heat equations with globally Lipschitz nonlinearity on a sequence of increasing domains, where the controls are acted on an equidistributed set that spreads out in the whole Euclidean space ℝN. As an application, we then show the exact null-controllability for this semilinear heat equation in ℝN. The main novelty here is that the upper bound on costs of null controls for such kind of equations in large but bounded domains can be made uniformly with respect to the sizes of domains under consideration. T

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6

Mahmudov, Nazim I. "Finite-approximate controllability of fractional evolution equations: variational approach." Fractional Calculus and Applied Analysis 21, no. 4 (2018): 919–36. http://dx.doi.org/10.1515/fca-2018-0050.

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Abstract In this work we extend a variational method to study the approximate controllability and finite dimensional exact controllability (finite-approximate controllability) for the fractional semilinear evolution equations with nonlocal conditions in Hilbert spaces. Assuming the approximate controllability of the corresponding linear equation we obtain sufficient conditions for the finite-approximate controllability of the fractional semilinear evolution equation under natural conditions. The obtained results are generalization and continuation of the recent results on this issue. Applicati

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7

Menezes, Silvano B. de, Juan Limaco, and Luis A. Medeiros. "Finite approximate controllability for semilinear heat equations in noncylindrical domains." Anais da Academia Brasileira de Ciências 76, no. 3 (2004): 475–87. http://dx.doi.org/10.1590/s0001-37652004000300002.

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We investigate finite approximate controllability for semilinear heat equation in noncylindrical domains. First we study the linearized problem and then by an application of the fixed point result of Leray-Schauder we obtain the finite approximate controllability for the semilinear state equation.

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8

Cesaroni, Annalisa, Nicolas Dirr, and Matteo Novaga. "Homogenization of a semilinear heat equation." Journal de l’École polytechnique — Mathématiques 4 (2017): 633–60. http://dx.doi.org/10.5802/jep.54.

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9

Ould Khatri, Mohamed Mahmoud, and Ahmed Youssfi. "Semilinear heat equation with singular terms." Electronic Journal of Qualitative Theory of Differential Equations, no. 69 (2022): 1–34. http://dx.doi.org/10.14232/ejqtde.2022.1.69.

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The main goal of this paper is to analyze the existence and nonexistence as well as the regularity of positive solutions for the following initial parabolic problem { ∂ t u − Δ u = μ u | x | 2 + f u σ in Ω T := Ω × ( 0 , T ) , u = 0 on ∂ Ω × ( 0 , T ) , u ( x , 0 ) = u 0 ( x ) in Ω , where Ω ⊂ R N , N ≥ 3 , is a bounded open, σ ≥ 0 and μ > 0 are real constants and f ∈ L m ( Ω T ) , m ≥ 1 , and u 0 are nonnegative functions. The study we lead shows that the existence of solutions depends on σ and the summability of the datum f as well as on the interplay between μ and the best constant in th

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10

Fabre, Caroline, Jean-Pierre Puel, and Enrike Zuazua. "Approximate controllability of the semilinear heat equation." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 125, no. 1 (1995): 31–61. http://dx.doi.org/10.1017/s0308210500030742.

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This article is concerned with the study of approximate controllability for the semilinear heat equation in a bounded domain Ω when the control acts on any open and nonempty subset of Ω or on a part of the boundary. In the case of both an internal and a boundary control, the approximate controllability in LP(Ω) for 1 ≦ p < + ∞ is proved when the nonlinearity is globally Lipschitz with a control in L∞. In the case of the interior control, we also prove approximate controllability in C0(Ω). The proof combines a variational approach to the controllability problem for linear equations and a fix

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11

Hu, Xiaomei. "Error Estimates for Solutions of the Semilinear Parabolic Equation in Whole Space." Abstract and Applied Analysis 2014 (2014): 1–5. http://dx.doi.org/10.1155/2014/501280.

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This paper is focused on the error estimates for solutions of the three-dimensional semilinear parabolic equation with initial datau0∈L2(ℝ3). Employing the energy methods and Fourier analysis technique, it is proved that the error between the solution of the semilinear parabolic equation and that of linear heat equation has the behavior asO((1+t)−3/8).

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12

Jacobe de Naurois, Ladislas, Arnulf Jentzen, and Timo Welti. "Weak Convergence Rates for Spatial Spectral Galerkin Approximations of Semilinear Stochastic Wave Equations with Multiplicative Noise." Applied Mathematics & Optimization 84, S2 (2021): 1187–217. http://dx.doi.org/10.1007/s00245-020-09744-6.

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AbstractStochastic wave equations appear in several models for evolutionary processes subject to random forces, such as the motion of a strand of DNA in a liquid or heat flow around a ring. Semilinear stochastic wave equations can typically not be solved explicitly, but the literature contains a number of results which show that numerical approximation processes converge with suitable rates of convergence to solutions of such equations. In the case of approximation results for strong convergence rates, semilinear stochastic wave equations with both additive or multiplicative noise have been co

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13

Chen, Yong Qing, and Hong Xu Li. "ASYMPTOTICALLY -ALMOST PERIODIC SOLUTIONS TO DIFFERENTIAL EQUATIONS IN BANACH SPACES." Far East Journal of Mathematical Sciences (FJMS) 141, no. 4 (2024): 299–316. http://dx.doi.org/10.17654/0972087124018.

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In this paper, we establish the existence and uniqueness result of asymptotically $(\omega, c)$-almost periodic mild solutions to semilinear differential equations in a Banach space. For this purpose, we first give some properties of $(\omega, c)$-almost periodic functions and asymptotically $(\omega, c)$-almost periodic functions, including the composition theorems. Then we obtain the existence and uniqueness result of $(\omega, c)$-almost periodic mild solution to the semilinear differential equation, and the existence and uniqueness theorems for $(\omega, c)$-almost periodic and asymptotica

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14

Fernandez-Cara, E. "Null controllability of the semilinear heat equation." ESAIM: Control, Optimisation and Calculus of Variations 2 (1997): 87–103. http://dx.doi.org/10.1051/cocv:1997104.

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15

Teresa, Luz de. "Insensitizing controls for a semilinear heat equation." Communications in Partial Differential Equations 25, no. 1-2 (2000): 39–72. http://dx.doi.org/10.1080/03605300008821507.

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16

Leiva, Hugo, N. Merentes, and J. Sanchez. "Approximate Controllability of a Semilinear Heat Equation." International Journal of Partial Differential Equations 2013 (November 3, 2013): 1–7. http://dx.doi.org/10.1155/2013/424309.

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We apply Rothe’s type fixed point theorem to prove the interior approximate controllability of the following semilinear heat equation: zt(t,x)=Δz(t,x)+1ωu(t,x)+f(t,z(t,x),u(t,x)) in (0,τ]×Ω,z=0, on (0,τ)×∂Ω,z(0,x)=z0(x), x∈Ω, where Ω is a bounded domain in ℝN (N≥1), z0∈L2(Ω), ω is an open nonempty subset of Ω, 1ω denotes the characteristic function of the set ω, the distributed control u belongs to L2(0,τ;L2(Ω)), and the nonlinear function f:[0,τ]×ℝ×ℝ→ℝ is smooth enough, and there are a,b,c∈ℝ, R>0 and 1/2≤β<1 such that |f(t,z,u)-az|≤c|u|β+b, for all u,z∈ ℝ,|u|,|z|≥R. Under this condition

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17

Huy Tuan, Nguyen, Daniel Lesnic, Tran Quoc Viet, and Vo Van Au. "Regularization of the semilinear sideways heat equation." IMA Journal of Applied Mathematics 84, no. 2 (2018): 258–91. http://dx.doi.org/10.1093/imamat/hxy058.

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18

Kwak, Minkyu. "A semilinear heat equation with singular initial data." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 128, no. 4 (1998): 745–58. http://dx.doi.org/10.1017/s0308210500021752.

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We first prove existence and uniqueness of non-negative solutions of the equationin in the range 1 < p < 1 + 2/N, when initial data u(x, 0) = a|x|−2(p−1), x ≠ 0, for a > 0. It is proved that the maximal and minimal solutions are self-similar with the formwhere g = ga satisfiesAfter uniqueness is proved, the asymptotic behaviour of solutions ofis studied. In particular, we show thatThe case for a = 0 is also considered and a sharp decay rate of the above equation is derived. In the final, we reveal existence of solutions of the first and third equations above, which change sign.

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19

Mahmudov, Nazim I. "Finite-Approximate Controllability of Riemann–Liouville Fractional Evolution Systems via Resolvent-Like Operators." Fractal and Fractional 5, no. 4 (2021): 199. http://dx.doi.org/10.3390/fractalfract5040199.

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This paper presents a variational method for studying approximate controllability and infinite-dimensional exact controllability (finite-approximate controllability) for Riemann–Liouville fractional linear/semilinear evolution equations in Hilbert spaces. A useful criterion for finite-approximate controllability of Riemann–Liouville fractional linear evolution equations is formulated in terms of resolvent-like operators. We also find that such a control provides finite-dimensional exact controllability in addition to the approximate controllability requirement. Assuming the finite-approximate

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20

Ascione, Giacomo, Daniele Castorina, Giovanni Catino, and Carlo Mantegazza. "A matrix Harnack inequality for semilinear heat equations." Mathematics in Engineering 5, no. 1 (2022): 1–15. http://dx.doi.org/10.3934/mine.2023003.

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<abstract><p>We derive a matrix version of Li &amp; Yau–type estimates for positive solutions of semilinear heat equations on Riemannian manifolds with nonnegative sectional curvatures and parallel Ricci tensor, similarly to what R. Hamilton did in <sup>[<xref ref-type="bibr" rid="b5">5</xref>]</sup> for the standard heat equation. We then apply these estimates to obtain some Harnack–type inequalities, which give local bounds on the solutions in terms of the geometric quantities involved.</p></abstract>

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21

Kaliuzhnyi-Verbovetskyi, Dmitry, and Georgi S. Medvedev. "The Semilinear Heat Equation on Sparse Random Graphs." SIAM Journal on Mathematical Analysis 49, no. 2 (2017): 1333–55. http://dx.doi.org/10.1137/16m1075831.

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22

Cho, Soyoung, and Minkyu Kwak. "SELF-SIMILAR SOLUTIONS OF A SEMILINEAR HEAT EQUATION." Taiwanese Journal of Mathematics 8, no. 1 (2004): 125–33. http://dx.doi.org/10.11650/twjm/1500558461.

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23

Guo, Jong-Shenq, and Bei Hu. "Quenching problem for a singular semilinear heat equation." Nonlinear Analysis: Theory, Methods & Applications 30, no. 2 (1997): 905–10. http://dx.doi.org/10.1016/s0362-546x(97)00357-x.

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24

Sebastian, Ani?a, and Tataru Daniel. "Null Controllability for the Dissipative Semilinear Heat Equation." Applied Mathematics and Optimization 46, no. 2 (2002): 97–105. http://dx.doi.org/10.1007/s00245-002-0746-2.

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25

Ishige, Kazuhiro, and Tatsuki Kawakami. "Refined asymptotic profiles for a semilinear heat equation." Mathematische Annalen 353, no. 1 (2011): 161–92. http://dx.doi.org/10.1007/s00208-011-0677-9.

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26

Liu, Weijiu, and Graham H. Williams. "Exact Internal Controllability for the Semilinear Heat Equation." Journal of Mathematical Analysis and Applications 211, no. 1 (1997): 258–72. http://dx.doi.org/10.1006/jmaa.1997.5459.

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27

Chouichi, Amel, and Sarah Otsmane. "The Cauchy problem for a system of nonlinear heat equations in two space dimensions." Asian-European Journal of Mathematics 08, no. 01 (2015): 1550004. http://dx.doi.org/10.1142/s1793557115500047.

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This paper is devoted to system of semilinear heat equations with exponential-growth nonlinearity in two-dimensional space which is the analogue of the scalar model problem studied in [S. Ibrahim, R. Jrad, M. Majdoub and T. Saanouni, Local well posedness of a 2D semilinear heat equation, Bull. Belg. Math. Soc.21 (2014) 1–17]. First, we prove the local existence and unconditional uniqueness of solutions in the Sobolev space (H1× H1)(ℝ2). The uniqueness part is nontrivial although it follows Brezis–Cazenave's proof [H. Brezis and T. Cazenave, A nonlinear heat equation with singular initial data,

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28

Leiva, Hugo, Miguel Narvaez, and Zoraida Sivoli. "Controllability of Impulsive Semilinear Stochastic Heat Equation with Delay." International Journal of Differential Equations 2020 (December 17, 2020): 1–10. http://dx.doi.org/10.1155/2020/2515160.

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LaSalle wrote the following: “it is never possible to start the system exactly in its equilibrium state, and the system is always subject to outside forces not taken into account by the differential equations. The system is disturbed and is displaced slightly from its equilibrium state. What happens? Does it remain near the equilibrium state? This is stability. Does it remain near the equilibrium state and in addition tend to return to the equilibrium? This is asymptotic stability.” Continuing with what LaSalle said, we conjecture that real-life systems are always under the influence of impuls

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29

Kalinichenko, Anatolii. "Mathematical Modelling of Processes of Spontaneous Ignition in a Stockpile with a Circular and Semicircular Section by Rothe’s and Two-Sided Approximations Methods." Mathematical and computer modelling. Series: Physical and mathematical sciences 26 (December 26, 2024): 37–49. https://doi.org/10.32626/2308-5878.2024-26.37-49.

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Self-ignition of a stockpile of materials such as peat, coal, and grain occurs due to the accumulation of heat released by an exothermic oxidation reaction, so the stockpile can be considered as a body with an internal heat source. The research of self-ignition processes using mathematical modeling is reduced to the need to find a solution to the initial boundary value problem for the two-dimensional semi-linear heat conduction equation. Since it is not always possible to find an analytical solution, it makes sense to use numerical analysis methods. The aim of this article is a numerical study

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Awadalla, Muath, Nazim I. Mahmudov та Jihan Alahmadi. "Finite-Approximate Controllability of ν-Caputo Fractional Systems". Fractal and Fractional 8, № 1 (2023): 21. http://dx.doi.org/10.3390/fractalfract8010021.

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This paper introduces a methodology for examining finite-approximate controllability in Hilbert spaces for linear/semilinear ν-Caputo fractional evolution equations. A novel criterion for achieving finite-approximate controllability in linear ν-Caputo fractional evolution equations is established, utilizing resolvent-like operators. Additionally, we identify a control strategy that not only satisfies the approximative controllability property but also ensures exact finite-dimensional controllability. Leveraging the approximative controllability of the corresponding linear ν-Caputo fractional e

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31

Galaktionov, V. A. "On blow-up and degeneracy for the semilinear heat equation with source." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 115, no. 1-2 (1990): 19–24. http://dx.doi.org/10.1017/s0308210500024537.

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SynopsisThe asymptotic behaviour of the solution of the semilinear parabolic equation ut = uxx + (1 + u)ln2(l + u) for t > 0, x ∊[−π, π ], ux(t, ± π) = 0 for t > 0 and u(0, x) = u0(x) ≧ 0 in [−π, π], which blows up at a finite time T0, is investigated. It is proved that for some two-parametric set of initial functions u0 the behaviour of u(t, x) near t = T0 is described by the approximate self-similar solution va(t, x) = exp {(T0 −t)−1 cos2 (x/2)} − 1, satisfying the first order nonlinear Hamilton–Jacobi equation vt, = (vx)2 /(1 + v) + (1 + v) ln2 (1 + v). Some open problems of degenerac

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32

Korman, Philip. "Necessary and sufficient condition for existence for a case of eigenvalues of multiplicity two." Electronic Journal of Differential Equations 2024, no. 01-?? (2024): 09. http://dx.doi.org/10.58997/ejde.2024.09.

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We establish necessary and sufficient condition for existence of solutions for a class of semilinear Dirichlet problems with the linear part at resonance at eigenvalues of multiplicity two. The result is applied to give a condition for unboundness of all solutions of the corresponding semilinear heat equation. For more information see https://ejde.math.txstate.edu/Volumes/2024/09/abstr.html

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33

Leiva, H., N. Merentes, and J. Sanchez. "A Characterization of Semilinear Dense Range Operators and Applications." Abstract and Applied Analysis 2013 (2013): 1–11. http://dx.doi.org/10.1155/2013/729093.

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We characterize a broad class of semilinear dense range operators given by the following formula, , where , are Hilbert spaces, , and is a suitable nonlinear operator. First, we give a necessary and sufficient condition for the linear operator to have dense range. Second, under some condition on the nonlinear term , we prove the following statement: If , then and for all there exists a sequence given by , such that . Finally, we apply this result to prove the approximate controllability of the following semilinear evolution equation: , where , are Hilbert spaces, is the infinitesimal generator

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34

An, Jing, Christopher Henderson, and Lenya Ryzhik. "Voting models and semilinear parabolic equations." Nonlinearity 36, no. 11 (2023): 6104–23. http://dx.doi.org/10.1088/1361-6544/ad001c.

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Abstract We present probabilistic interpretations of solutions to semi-linear parabolic equations with polynomial nonlinearities in terms of the voting models on the genealogical trees of branching Brownian motion (BBM). These extend McKean’s connection between the Fisher–KPP equation and BBM (McKean 1975 Commun. Pure Appl. Math. 28 323–31). In particular, we present ‘random outcome’ and ‘random threshold’ voting models that yield any polynomial nonlinearity f satisfying f ( 0 ) = f ( 1 ) = 0 and a ‘recursive up the tree’ model that allows to go beyond this restriction on f. We compute several

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35

HILLEN, T. "QUALITATIVE ANALYSIS OF SEMILINEAR CATTANEO EQUATIONS." Mathematical Models and Methods in Applied Sciences 08, no. 03 (1998): 507–19. http://dx.doi.org/10.1142/s0218202598000238.

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The linear Cattaneo equation appears in heat transport theory to describe heat wave propagation with finite speed. It can also be seen as a generalization of a correlated random walk. If the system admits nonconservative forces (or reactions), then a nonlinear Cattaneo system is obtained. Here we consider asymptotic behavior of solutions of the nonlinear Cattaneo system. Following Brayton and Miranker we define a Lyapunov function to show global existence of solutions and to show that each ω-limit set is contained in the set of all stationary solutions.

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36

WU, HUI. "A FUJITA-TYPE RESULT FOR A SEMILINEAR EQUATION IN HYPERBOLIC SPACE." Journal of the Australian Mathematical Society 103, no. 3 (2017): 420–29. http://dx.doi.org/10.1017/s1446788717000052.

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In this paper, we study the positive solutions for a semilinear equation in hyperbolic space. Using the heat semigroup and by constructing subsolutions and supersolutions, a Fujita-type result is established.

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Teresa, Luz de, and Enrique Zuazua. "Null controllability of linear and semilinear heat equations in thin domains." Asymptotic Analysis 24, no. 3-4 (2000): 295–317. https://doi.org/10.3233/asy-2000-418.

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We consider the linear heat equation with potential in a n‐dimensional thin cilinder Ωε=Ω×(0,ε) where Ω is a bounded open smooth set of $\mathbb{R}^{n-1}$ with n≥2 and ε is a small parameter. We study the null controllability problem when the control acts in a cylindrical region ωε=ω×(0,ε), where ω⊂Ω is an open and non‐empty subset of Ω. We prove that, under appropriate boundary conditions, for a suitable class of potentials the heat equation is uniformly null controllable as ε→0. We also prove the convergence of the controls to a null control for the n−1‐dimensional heat equation in Ω. Simila

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38

Fahrenwaldt, Matthias A. "Short-time asymptotic expansions of semilinear evolution equations." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 146, no. 1 (2016): 141–67. http://dx.doi.org/10.1017/s0308210515000372.

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We develop an algebraic approach to constructing short-time asymptotic expansions of solutions of a class of abstract semilinear evolution equations. The expansions are typically valid for both the solution of the equation and its gradient. We apply a perturbation approach based on the symbolic calculus of pseudo-differential operators and heat kernel methods. The construction is explicit and can be done to arbitrary order. All results are rigorously formulated in terms of Banach algebras. As an application we obtain a novel approach to finding approximate solutions of Markovian backward stoch

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39

Wang, Lijuan. "Minimal Time Impulse Control Problem of Semilinear Heat Equation." Journal of Optimization Theory and Applications 188, no. 3 (2021): 805–22. http://dx.doi.org/10.1007/s10957-020-01807-6.

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Ibrahim, Slim, Rym Jrad, Mohamed Majdoub, and Tarek Saanouni. "Local well posedness of a 2D semilinear heat equation." Bulletin of the Belgian Mathematical Society - Simon Stevin 21, no. 3 (2014): 535–51. http://dx.doi.org/10.36045/bbms/1407765888.

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Zuazua, Enrique. "Finite dimensional null controllability for the semilinear heat equation." Journal de Mathématiques Pures et Appliquées 76, no. 3 (1997): 237–64. http://dx.doi.org/10.1016/s0021-7824(97)89951-5.

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42

Gianni, Roberto, and Josephus Hulshof. "The semilinear heat equation with a Heaviside source term." European Journal of Applied Mathematics 3, no. 4 (1992): 367–79. http://dx.doi.org/10.1017/s0956792500000917.

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We consider the initial value problem for the equation ut = uxx + H(u), where H is the Heaviside graph, on a bounded interval with Dirichlet boundary conditions, and discuss existence, regularity and uniqueness of solutions and interfaces.

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43

Wang, Zhiyong, and Jingxue Yin. "A note on semilinear heat equation in hyperbolic space." Journal of Differential Equations 256, no. 3 (2014): 1151–56. http://dx.doi.org/10.1016/j.jde.2013.10.011.

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44

Harada, Junichi. "Blowup profile for a complex valued semilinear heat equation." Journal of Functional Analysis 270, no. 11 (2016): 4213–55. http://dx.doi.org/10.1016/j.jfa.2016.03.015.

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45

Mizoguchi, Noriko. "Multiple blowup of solutions for a semilinear heat equation." Mathematische Annalen 331, no. 2 (2004): 461–73. http://dx.doi.org/10.1007/s00208-004-0590-6.

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Harada, Junichi. "Nonsimultaneous blowup for a complex valued semilinear heat equation." Journal of Differential Equations 263, no. 8 (2017): 4503–16. http://dx.doi.org/10.1016/j.jde.2017.05.024.

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47

Zygouras, Nikos. "Exponential convergence for a periodically driven semilinear heat equation." Annales de l'Institut Henri Poincare (C) Non Linear Analysis 26, no. 1 (2009): 271–84. http://dx.doi.org/10.1016/j.anihpc.2008.01.003.

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48

Taira, Kazuaki. "Semilinear elliptic boundary-value problems in combustion theory." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 132, no. 6 (2002): 1453–76. http://dx.doi.org/10.1017/s0308210500002201.

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This paper is devoted to the study of semilinear degenerate elliptic boundary-value problems arising in combustion theory that obey a general Arrhenius equation and a general Newton law of heat exchange. We prove that ignition and extinction phenomena occur in the stable steady temperature profile at some critical values of a dimensionless rate of heat production.

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49

Lemoine, Jérôme, Irene Marín-Gayte, and Arnaud Münch. "Approximation of null controls for semilinear heat equations using a least-squares approach." ESAIM: Control, Optimisation and Calculus of Variations 27 (2021): 63. http://dx.doi.org/10.1051/cocv/2021062.

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The null distributed controllability of the semilinear heat equation ∂ty − Δy + g(y) = f 1ω assuming that g ∈ C1(ℝ) satisfies the growth condition lim sup|r|→∞g(r)∕(|r|ln3∕2|r|) = 0 has been obtained by Fernández-Cara and Zuazua (2000). The proof based on a non constructive fixed point theorem makes use of precise estimates of the observability constant for a linearized heat equation. Assuming that g′ is bounded and uniformly Hölder continuous on ℝ with exponent p ∈ (0, 1], we design a constructive proof yielding an explicit sequence converging strongly to a controlled solution for the semilin

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50

Anukiruthika, K., N. Durga, and P. Muthukumar. "Approximate controllability of semilinear retarded stochastic differential system with non-instantaneous impulses: Fredholm theory approach." IMA Journal of Mathematical Control and Information 38, no. 2 (2021): 684–713. http://dx.doi.org/10.1093/imamci/dnab006.

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Abstract This article deals with the approximate controllability of semilinear retarded integrodifferential equations with non-instantaneous impulses governed by Poisson jumps in Hilbert space. The existence of a mild solution is established by using stochastic calculus and a suitable fixed point technique. The approximate controllability of the proposed non-linear stochastic differential system is obtained by employing the theory of interpolation spaces and Fredholm theory. Finally, applications to the stochastic heat equation and retarded type stochastic Benjamin–Bona–Mahony equation are pro

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