**Approximate controllability of the semilinear heat equation**

This article cares with the study of approximate managementlability for the semilinear heat equation in a very delimited domain Ω once the control acts on any open and nonempty set of Ω or on a section of the boundary. within the case of each an enclosed and a boundary management, the approximate controllability in LP(Ω) for one ≦ p < + ∞ is verified once the nonlinearity is globally Lipschitz with an impact in L∞. within the case of the inside management, we have a tendency to additionally prove approximate controllability in C0(Ω). The proof combines a variational approach to the controllability drawback for linear equations and a hard and fast purpose methodology. we have a tendency to additionally prove that the management may be taken to be of “quasi bang-bang” kind.** [1]**

**Existence and non-existence of global solutions for a semilinear heat equation**

The existence and non-existence of worldwide solutions and theL p blow-up of non-global solutions to the initial price problemu′(t)=Δu(t)+u(t) γ onR n are studied. we have a tendency to contemplate onlyγ>1. within the casen(γ − 1)/2=1, we have a tendency to gift a straightforward proof that there are not any non-trivial international non-negative solutions. Ifn(γ−1)/2≦1, we have a tendency to show underneath delicate technical restrictions that non-negativeL p solutions continually blow-up inL p norm in finite time. within the casen(γ−1)/2>1, we have a tendency to offer new spare conditions on the initial information that guarantee the existence of worldwide solutions. **[2]**

**Large fluctuations for a nonlinear heat equation with noise**

Studies a nonlinear heat equation in an exceedingly finite interval of area subject to a dissonance forcing term. The equation while not the forcing term exhibits many equilibrium configurations, 2 of that are stable. the answer of the entire forced equation could be a model in area and time that incorporates a distinctive random equilibrium. The authors study this method within the limit of little noise, and acquire lower and higher bounds for the chance of enormous fluctuations. They then apply these estimates to calculate the transition chance between the stable configurations (tunnelling). This model drawback will be taken as a rigorous version of some recent tries to explain geometer quantum systems in terms of random equilibrium states of a nonlinear stochastic equation in infinite dimensions.** [3]**

**A transient thermal cloak experimentally realized through a rescaled diffusion equation with anisotropic thermal diffusivity**

Transformation optics has created a significant contribution to the advancement of contemporary electromagnetism and connected analysis motor-assisted by the event of metamaterials. during this work, we tend to applied this idea to the physical science mistreatment the coordinate transformation to the time-dependent heat diffusion equation to govern the warmth flux by predefined diffusion methods. by experimentation, we tend to incontestible a transient thermal cloaking device built with effective thermal materials and with success hid a centimeter-sized vacuum cavity. A rescaled heat equation accounting for all the pertinent parameters of varied ingredient materials was projected to greatly facilitate the fabrication. **[4]**

**Solving Fuzzy Heat Equation by Using Numerical Methods**

This analysis proposes a particular methodology to unravel fuzzy heat equation with integral boundary conditions. the required materials and preliminaries area unit expressed, and a finite distinction theme for one dimensional heat equation is taken into account. Here, boundary conditions embody integral equations that area unit approximated by the composite trapezoid rule. Finally, associate example so as as an example the numerical results is given. during this example, the Hausdor distance between actual answer and approximate answer is obtained. **[5]**

**Reference**

**[1]** Fabre, C., Puel, J.P. and Zuazua, E., 1995. Approximate controllability of the semilinear heat equation. Proceedings of the Royal Society of Edinburgh Section A: Mathematics, 125(1), (Web Link)

**[2]** Weissler, F.B., 1981. Existence and non-existence of global solutions for a semilinear heat equation. Israel Journal of Mathematics, 38(1-2), (Web Link)

**[3]** Faris, W.G. and Jona-Lasinio, G., 1982. Large fluctuations for a nonlinear heat equation with noise. Journal of Physics A: Mathematical and General, 15(10), (Web Link)

**[4]** A transient thermal cloak experimentally realized through a rescaled diffusion equation with anisotropic thermal diffusivity

Yungui Ma, Lu Lan, Wei Jiang, Fei Sun & Sailing He

NPG Asia Materialsvolume 5, pagee73 (2013) (Web Link)

**[5]** Hosseinpour, A. (2018) “Solving Fuzzy Heat Equation by Using Numerical Methods”, Asian Research Journal of Mathematics, 8(1), (Web Link)