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Talk 01 — Prof. Phuc Cong Nguyen (Louisiana State University, USA) — [computing…]Local time: click here to view in your time zoneTitle: Existence criteria for BMO solutions to quasilinear equations with measure data
Abstract
We give sharp criteria for the existence of BMO solutions to the quasilinear equation $-\Delta_{p} u = \sigma u^{q} + \mu$ in $\mathbb{R}^n$, $u\ge 0$, with $q>0$, where both $\mu$ and $\sigma$ are locally finite Radon measures, and $\Delta_{p}u= \text{div}(|\nabla u|^{p-2}\nabla u)$ is the $p$-Laplacian ($p>1$). Our results also hold for a class of more general quasilinear operators ${\rm div}\, (\mathcal{A}(x, \nabla \cdot))$ in place of $\Delta_{p}$. This talk is based on joint work with Igor E. Verbitsky.
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Talk 02 — Prof. S. Rajesh (Indian Institute of Technology Tirupathi, India) — [computing…]Local time: click here to view in your time zoneTitle: Fixed point results for α-nonexpansive mappings
Abstract
A Banach space X is said to have the weak fixed point property if every nonexpansive T map defined on a nonempty weakly compact and convex set K has a fixed point in K. It is known that Hilbert spaces and uniformly convex Banach spaces have the fixed point property. Moreover, if a Banach space X has the geometric property normal structure, then it has the fixed point property. Aoyama and Kohsaka introduced α-nonexpansive mappings and proved the existence of fixed points for such mappings defined on weakly compact convex subsets of Hilbert spaces. Note that if a map is nonexpansive, then it is α-nonexpansive. But the converse need not hold, as α-nonexpansive maps need not be continuous. In this talk, we will show that if a Banach space X is Orthogonally convex, then X has the fixed-point property for α-nonexpansive mappings. This is a joint work with Professor Enrique Llorens-Fuster, Department of Mathematical Analysis, University of Valencia, Valencia, Spain.
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Talk 03 — Prof. Satyanarayana Engu (National Institute of Technology Warangal, India) — [computing…]Local time: click here to view in your time zoneTitle: Large-time asymptotics of the solutions to a KdV–Burgers equation
Abstract
We consider the initial value problem for a KDV-Burgers equation. We establish the global existence of the solutions and study its asymptotic behavior. For which, we construct the asymptotic profiles and do the asymptotic analysis for large time.
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Talk 04 — Prof. Tanmay Sarkar (Indian Institute of Technology Jammu, India) — [computing…]Local time: click here to view in your time zoneTitle: Stable determination of potential in wave equation
Abstract
In this talk, we consider time-dependent matrix potential in a system of wave equation. Our objective is to provide a stability estimate with respect to input-output map.
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Talk 05 — Prof. Naga Raju Gande (Visvesvaraya National Institute of Technology Nagpur, India) — [computing…]Local time: click here to view in your time zoneTitle: From Heaviside to Heat Kernel: Constructing the Fundamental Solution via Special Cauchy Data
Abstract
The heat kernel, or fundamental solution of the diffusion equation, is essential for solving initial value problems in heat propagation and diffusion processes. This talk presents an elegant constructive approach to deriving the heat kernel by strategically choosing special initial conditions: the Heaviside step function. We demonstrate how solving the heat equation with this particular step function initial data exploits the self-similarity and scaling symmetry inherent in the problem. By seeking solutions that remain invariant under appropriate scaling transformations, the partial differential equation reduces to a simpler ordinary differential equation. The fundamental solution then emerges naturally by taking the spatial derivative of this special solution, revealing the classical Gaussian heat kernel. This constructive method illuminates the profound connection between symmetry principles, judicious choice of initial conditions, and the general theory of heat propagation. The approach extends naturally to higher spatial dimensions and provides insight into why the heat kernel takes its characteristic form.
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Talk 06 — Prof. Triveni Prasad Shukla (National Institute of Technology Warangal, India) — [computing…]Local time: click here to view in your time zoneTitle: Dynamics of shock waves in non-ideal fluids exhibiting quartic nonlinearity
Abstract
This research provides a detailed analysis of shock wave propagation in media exhibiting mixed-type weak nonlinearity. The study employs the one-dimensional Euler equations, supplemented by the van der Waals equation of state. An undisturbed state is considered in which weak nonlinearity causes the fundamental derivative to change sign twice, resulting in quartic nonlinearity. This phenomenon produces both expansion and compression waves within a single pulse. A comprehensive analytical investigation of wave interactions reveals several notable and unconventional outcomes. These include the formation of multiple precursor waves that extinguish in finite time, the emergence of a centered wave fan during interactions, and the splitting of a single shock wave into two distinct shock waves, phenomena not observed in the absence of quartic nonlinearity. Furthermore, the influence of non-ideal fluid parameters on interaction times and overall wave propagation is examined.
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Talk 07 — Prof. Subhankar Mondal (National Institute of Technology Warangal, India) — [computing…]Local time: click here to view in your time zoneTitle: An inverse identification problem for bi-parabolic equation
Abstract
We consider an inverse problem of identification of a spatial source function from final time observation associated with a bi-parabolic equation. We shall show that the considered problem can be modeled as solving a linear operator equation, where the modelling operator is a compact operator of infinite rank and hence implying that the problem is ill-posed. Thus, a regularization method has to be employed in order to obtain some stable approximations. For this, we shall make use of the quasi-reversibility method and analyse the convergence rates. Under some standard source conditions we shall obtain the H{\"o}lder rate of convergence for the apriori and aposteriori parameter choice strategies which improve upon the rates obtained in earlier works. If time permits, we shall also discuss about the optimality of the obtained rates.
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Talk 08 — Prof. Manmohan Vashisth (Indian Institute of Technology Ropar, India) — [computing…]Local time: click here to view in your time zoneTitle: Calderon type inverse problems for linear and nonlinear partial differential equations
Abstract
In this talk, we discuss our findings on Calderón-type inverse problems for both linear and nonlinear hyperbolic partial differential equations (PDEs). We focus on the reconstruction of coefficients associated with both linear and nonlinear terms in semilinear hyperbolic PDEs from boundary measurements, specifically using the Dirichlet-to-Neumann map.
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Talk 09 — Prof. Mohan Kumar Mallick (Visvesvaraya National Institute of Technology Nagpur, India) — [computing…]Local time: click here to view in your time zoneTitle: Symmetry and Shape Optimization for the Principal Eigenvalue of Pucci’s Supremum Operator
Abstract
Shape optimization of principal eigenvalues is a central theme in spectral geometry and elliptic partial differential equations. Classical results such as the Faber–Krahn inequality and the Pólya–Szegő conjecture suggest that symmetry of the domain tends to minimize the first Dirichlet eigenvalue of the Laplace operator. However, for fully nonlinear elliptic operators, especially Pucci’s extremal operators, the absence of a variational characterization makes the corresponding shape optimization problems substantially more difficult. In this talk, we discuss an explicit example showing that symmetry minimizes the positive principal eigenvalue for Pucci’s supremum operator within a special class of planar domains. After recalling the notion of principal eigenvalue in the fully nonlinear setting through the maximum principle, we describe a family of domains depending on the ellipticity ratio and geometric deformation parameters. For these domains, the principal eigenvalue and the associated positive eigenfunction can be computed explicitly. The construction shows that, among domains of fixed area in this class, the minimum of the principal eigenvalue is achieved by the most symmetric member of the family. The talk highlights how this example provides a fully nonlinear analogue of the symmetry-minimization philosophy behind classical spectral inequalities. It also illustrates the technical challenges caused by the non-variational nature of Pucci operators and points toward broader open questions concerning Faber–Krahn type inequalities and Pólya–Szegő type conjectures for fully nonlinear elliptic operators.
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Talk 10 — Prof. Sushmita Rawat (National Institute of Technology Warangal, India) — [computing…]Local time: click here to view in your time zoneTitle: Exploring nonlocal elliptic problems with Choquard nonlinearity
Abstract
In this talk, we investigate the existence of weak solutions for elliptic problems involving Choquard nonlinearity. These equations have attracted significant attention due to their ability to model long-range in- teractions in various real-world applications. A key concept in solving PDEs is that of weak solutions. These solutions satisfy the integral form of the PDE and are useful when classical solutions may not exist or are challenging to compute. We will use Variational methods to solve the PDEs. This technique essen- tially transforms the problem of solving a PDE into the problem of finding critical points of the associated functional.